Molecular Characterization of Hardy Spaces Associated with Twisted Convolution

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.

1. Introduction

In this paper, we consider the 2𝑛 linear differential operators 𝑍_𝑗=𝜕𝜕𝑧_𝑗+14𝑧_𝑗,𝑍_𝑗=𝜕𝜕𝑧_𝑗−14𝑧_𝑗,onℂ^𝑛,𝑗=1,2,…,𝑛. (1) Together with the identity they generate a Lie algebra ℎ^𝑛 which is isomorphic to the 2𝑛+1 dimensional Heisenberg algebra. The only nontrivial commutation relations are 𝑍_𝑗,𝑍_𝑗1=−2𝐼,𝑗=1,2,…,𝑛. (2) The operator 𝐿 defined by 1𝐿=−2^𝑛_𝑗=1𝑍_𝑗𝑍_𝑗+𝑍_𝑗𝑍_𝑗 (3) is nonnegative, self-adjoint, and elliptic. Therefore, it generates a diffusion semigroup {𝑇^𝐿_𝑡}_𝑡>0={𝑒^−𝑡𝐿}_𝑡>0. The operators in (1) generate a family of “twisted translations” 𝜏_𝑤 on ℂ^𝑛 defined on measurable functions by 𝜏_𝑤𝑓(𝑧)1=exp2^𝑛_𝑗=1𝑤_𝑗𝑧_𝑗+𝑤_𝑗𝑧_𝑗𝑓(𝑧)()𝑖=𝑓𝑧+𝑤exp2Im(𝑧⋅𝑠𝑤). (4) The “twisted convolution” of two functions 𝑓 and 𝑔 on ℂ^𝑛 can now be defined as 𝑓×𝑔(𝑧)=_ℂ𝑛𝑓(𝑤)𝜏_−𝑤𝑔(𝑧)=𝑑𝑤_ℂ𝑛𝑓(𝑧−𝑤)𝑔(𝑤)𝜔(𝑧,𝑤)𝑑𝑤, (5) 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 𝐻¹_𝐿(ℂ^𝑛) associated with twisted convolution. They gave several characterizations of 𝐻¹_𝐿(ℂ^𝑛) 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 𝐻^𝑝_𝐿(ℂ^𝑛) 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 𝐻^𝑝_𝐿(ℂ^𝑛). As an application, we prove the boundedness of the local Riesz transform on the Hardy space 𝐻^𝑝_𝐿(ℂ^𝑛).

We first give some basic notations about 𝐻^𝑝_𝐿(ℂ^𝑛). Let ℬ denote the class of 𝐶^∞-functions 𝜑 on ℂ^𝑛, supported on the ball 𝐵(0,1) such that ‖𝜑‖_∞≤1 and ‖∇𝜑‖_∞≤2. For 𝑡>0, let 𝜑_𝑡(𝑧)=𝑡^−2𝑛𝜑(𝑧/𝑡). Given 𝜎>0, 0<𝜎≤+∞, and a tempered distribution 𝑓, define the grand maximal function 𝑀_𝜎𝑓(𝑧)=sup_𝜑∈ℬsup_0<𝑡<𝜎𝜑_𝑡×𝑓(𝑧). (6) Then, the Hardy space 𝐻^𝑝_𝐿(ℂ^𝑛) can be defined by 𝐻^𝑝_𝐿ℂ^𝑛=𝑓∈𝒮^′ℂ^𝑛∶𝑀_∞𝑓∈𝐿^𝑝ℂ^𝑛. (7) For any 𝑓∈𝐻^𝑝_𝐿(ℂ^𝑛), define ‖𝑓‖_𝐻^𝑝_{𝐿(ℂ𝑛)}=‖𝑀_∞𝑓‖_𝐿𝑝.

Definition 1. Let 0<𝑝≤1≤𝑞≤∞ and 𝑝≠𝑞. A function 𝑎(𝑧) is a 𝐻_𝐿^𝑝,𝑞-atom for the Hardy space 𝐻^𝑝_𝐿(ℂ^𝑛) associated to a ball 𝐵(𝑧₀,𝑟) if (1)supp𝑎⊂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 space 𝐻_𝐿^𝑝,𝑞(ℂ^𝑛) to be the set of all tempered distributions of the form ∑_𝑗𝜆_𝑗𝑎_𝑗 (the sum converges in the topology of 𝒮^′(ℂ^𝑛)), where 𝑎_𝑗 are 𝐻_𝐿^𝑝,𝑞-atoms and ∑_𝑗|𝜆_𝑗|^𝑝<+∞.

The atomic quasinorm in 𝐻_𝐿^𝑝,𝑞(ℂ^𝑛) is defined by 𝑓_𝐿-atom=inf_𝑗𝜆_𝑗^𝑝1/𝑝, (8) where the infimum is taken over all decompositions 𝑓=∑_𝑗𝜆_𝑗𝑎_𝑗 and 𝑎_𝑗 are 𝐻_𝐿^𝑝,𝑞-atoms.

The following result has been proved in [4, 7].

Proposition 2. Let 2𝑛/(2𝑛+1)<𝑝≤1. Then, for a tempered distribution 𝑓 on ℂ^𝑛, the following are equivalent:(i)𝑀_∞𝑓∈𝐿^𝑝(ℂ^𝑛);(ii)for some 𝜎, 0<𝜎<+∞, 𝑀_𝜎𝑓∈𝐿^𝑝(ℂ^𝑛);(iii)for some radial function 𝜑∈𝒮, such that ∫_ℂ𝑛𝜑(𝑧)𝑑𝑧≠0, we have sup_0<𝑡<1𝜑_𝑡×𝑓(𝑧)∈𝐿^𝑝ℂ^𝑛; (9)(iv)𝑓 can be decomposed as 𝑓=∑_𝑗𝜆_𝑗𝑎_𝑗, where 𝑎_𝑗 are 𝐻_𝐿^𝑝,𝑞-atoms and ∑_𝑗|𝜆_𝑗|^𝑝<+∞.

Corollary 3. Let 2𝑛/(2𝑛+1)<𝑝≤1 and 1<𝑞≤∞. Then, 𝐻_𝐿^𝑝,𝑞(ℂ^𝑛)=𝐻^𝑝_𝐿(ℂ^𝑛) with equivalent norms.

In order to give the main result of this paper, we need the dual space of Hardy space 𝐻^𝑝_𝐿(ℂ^𝑛).

Definition 4. Let 0≤𝛼<1/2𝑛; a locally integrable function 𝑓 is said to be in the Campanato type space Λ^𝐿_𝛼 if there exists a constant 𝐾>0 such that, for every ball 𝐵=𝐵(𝑧,𝑟), |𝐵|^−𝛼_𝐵𝑓(𝜐)−1|𝐵|_𝐵𝑓(𝑢)𝜔(𝑧,𝑢)𝑑𝑢𝜔(𝑧,𝜐)²×𝑑𝜐|𝐵|^1/2≤𝐾. (10) The norm ‖𝑓‖_Λ𝐿𝛼 of 𝑓 is the least value of 𝐾 for which the above inequality holds.

The dual space of 𝐻¹_𝐿(ℂ^𝑛) is the BMO type space BMO_𝐿(ℂ^𝑛) (cf. [4]). Note that Λ^𝐿₀ is identified with BMO_𝐿. Let ℋ_𝐿^𝑝,2,𝑎 denote the space of finite linear combinations of 𝐻_𝐿^𝑝,2-atoms, which coincides with 𝐿²_𝑐(ℂ^𝑛), the space of square integrable functions with compact support. By Proposition 2, ℋ_𝐿^𝑝,2,𝑎 is a dense subspace of 𝐻^𝑝_𝐿(ℂ^𝑛). Set ℒ_𝑔𝑓=_ℂ𝑛𝑓(𝑧)𝑔(𝑧)𝑑𝑧,𝑓∈ℋ_𝐿^𝑝,2,𝑎,𝑔∈𝐿²_locℂ^𝑛. (11)

Similar to the classical case in [10], we immediately obtain the following theorem which proves that Λ^𝐿_1/𝑝−1 is the dual space of 𝐻^𝑝_𝐿(ℂ^𝑛) for 2𝑛/(2𝑛+1)<𝑝<1.

Theorem 5. Let 2𝑛/(2𝑛+1)<𝑝<1. Then(a)suppose 𝑔∈Λ^𝐿_(1/𝑝)−1; then ℒ_𝑔 given by (11) extends to a bounded linear functional on 𝐻^𝑝_𝐿(ℂ^𝑛) and satisfies ℒ_𝑔≤𝐶𝑔_{Λ𝐿(1/𝑝)−1}; (12)(b)conversely, every bounded linear functional ℒ on 𝐻^𝑝_𝐿(ℂ^𝑛) can be realized as ℒ=ℒ_𝑔 with 𝑔∈Λ^𝐿_(1/𝑝)−1 and 𝑔_{Λ𝐿(1/𝑝)−1}≤𝐶‖ℒ‖. (13)

Remark 6. We may define the space Λ^𝐿_{(1/𝑝)−1,𝑞′},2𝑛/(2𝑛+1)<𝑝<1, 1≤𝑞^′≤∞, by |𝐵|^1−(1/𝑝)_𝐵𝑓(𝜐)−1|𝐵|_𝐵𝑓(𝑢)𝜔(𝑧,𝑢)𝑑𝑢𝜔(𝑧,𝜐)^𝑞′×𝑑𝜐|𝐵|^1/𝑞′≤𝐾, (14) where 𝐵=𝐵(𝑧,𝑟). The norm ‖𝑓‖_{Λ𝐿′(1/𝑝)−1,𝑞} of 𝑓 is the least value of 𝐾 for which the above inequality holds. Due to Theorem 5, Λ^𝐿_{(1/𝑝)−1,𝑞′} is also identified with the dual space of 𝐻^𝑝_𝐿(ℂ^𝑛). The proof is almost the same as that of Theorem 5. Thus, the space Λ^𝐿_{(1/𝑝)−1,𝑞′} coincides with Λ^𝐿_(1/𝑝)−1 and ‖𝑓‖_{Λ𝐿′(1/𝑝)−1,𝑞}∼‖𝑓‖_{Λ𝐿(1/𝑝)−1}.

Definition 7. Let 2𝑛/(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∈𝐿^𝑞 is called a 𝐻_𝐿^{𝑝,𝑞,𝜖}-molecule with the center 𝑧₀ if (1)|𝑧|^2𝑛𝑏𝑀(𝑧)∈𝐿^𝑞, (2)𝒩(𝑀)=‖𝑀‖_𝐿^𝑎/𝑏_𝑞‖|⋅−𝑧₀|^2𝑛𝑏𝑀‖_𝐿^{1−(𝑎/𝑏)}_𝑞<∞, (3)∫_ℂ𝑛𝑀(𝑧)𝜔(𝑧₀,𝑧)𝑑𝑧=0.
Then, we can obtain a molecular characterization of 𝐻^𝑝_𝐿(ℂ^𝑛) as follows.

Theorem 8. Given 𝑝,𝑞,𝜖 as in Definition 7, then 𝑓∈𝐻^𝑝_𝐿 if and only if 𝑓 can be written as 𝑓=∑_𝑗𝜆_𝑗𝑀_𝑗, where 𝑀_𝑗 are 𝐻_𝐿^{𝑝,𝑞,𝜖}-molecules and ∑_𝑗|𝜆_𝑗|^𝑝<∞. The sum converges in 𝐻^𝑝_𝐿 norm and also in (Λ^𝐿_(1/𝑝)−1)^∗ when 2𝑛/(2𝑛+1)<𝑝<1. Moreover, 𝑓_𝐻^𝑝_𝐿∼𝑓_𝐻𝐿^{𝑝,𝑞,𝜖,𝑀}=inf_𝑗𝜆_𝑗^𝑝1/𝑝, (15) where the infimum is taken over all decompositions of 𝑓 into 𝐻_𝐿^{𝑝,𝑞,𝜖}-molecules.

Let 𝜓 be a 𝐶^∞-function on ℂ^𝑛 with compact support and such that 𝜓≡1 on a neighborhood of zero. Define 𝑅_𝑗(𝑧)=𝑧_𝑗|𝑧|^2𝑛+1𝜓(𝑧),𝑅_𝑗(𝑧)=𝑧_𝑗|𝑧|^2𝑛+1𝜓(𝑧), (16) for 𝑗=1,2,…,𝑛.

We refer to the singular integral operators 𝑅_𝑗,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 𝑍_𝑗𝐿^−1/2, 𝑍_𝑗𝐿^−1/2, 𝑗=1,2,…,𝑛.

As an application of Theorem 8, we can prove the following.

Theorem 9. The local Riesz transforms 𝑅_𝑗,𝑅_𝑗, 𝑗=1,2,…,𝑛 are bounded on 𝐻^𝑝_𝐿(ℂ^𝑛), where 2𝑛/(2𝑛+1)<𝑝≤1.

Remark 10. When 𝑝=1, Theorem 9 is proved by the connection between 𝐻¹_𝐿(ℂ^𝑛) and Hardy space on the Heisenberg group 𝐻¹(ℍ_𝑛) (cf. Lemma  4.9 in [4]).

Throughout the paper, 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.

2. Molecule Characterization of 𝐻^𝑝_𝐿(ℂ^𝑛)

In this section, we prove the main result of this paper. Firstly, we have the following lemma.

Lemma 11. If 𝑎 is a 𝐻_𝐿^𝑝,𝑞-atom for 2𝑛/(2𝑛+1)<𝑝≤1≤𝑞≤+∞ supported in 𝐵(𝑧₀,𝑟), then 𝑎 is a 𝐻_𝐿^{𝑝,𝑞,𝜀}-molecule centered at 𝑧₀ and 𝒩(𝑎)≤𝐶, (17) where 𝜖>0 and 𝐶 is a positive constant that is independent of 𝑎.

Proof. Since ‖𝑎‖_𝑞≤|𝐵|^{1/𝑞−1/𝑝}=|𝐵|^𝑎−𝑏, (18) we get ⋅−𝑧₀^2𝑛𝑏𝑎(⋅)_𝑞≤𝑟^2𝑛𝑏‖𝑎‖_𝑞≤𝐶|𝐵|^𝑏|𝐵|^𝑎−𝑏=𝐶|𝐵|^𝑎. (19) Therefore, 𝒩(𝑎)≤𝐶|𝐵|^{(𝑎/𝑏)(𝑎−𝑏)}|𝐵|^{𝑎(1−(𝑎/𝑏))}=𝐶. (20) This proves that 𝑎 is a molecule with center at 𝑧₀.

The following lemma is the key step for the proof of Theorem 8.

Lemma 12. If 𝑀 is a 𝐻_𝐿^{𝑝,𝑞,𝜖}-molecule with center at 𝑧₀, then 𝑀∈𝐻_𝐿^𝑝,𝑞(ℂ^𝑛) and ‖𝑀‖_𝐻^𝑝_𝐿≤𝐶𝒩(𝑀), (21) where 𝐶 is independent of 𝑀.

Proof. If 𝑞=2, let 𝜎=‖𝑀‖₂^{1/(2𝑛(𝑎−𝑏))}, 𝐸₀={𝑧∈ℂ^𝑛∶|𝑧−𝑧₀|≤𝜎}, and 𝐸_𝑘={𝑧∈ℂ^𝑛∶2^𝑘−1𝜎<|𝑧−𝑧₀|≤2^𝑘𝜎}. Denote 𝑀_𝑘=𝑀𝜒_𝑘, where 𝜒_𝑘 is the characteristic function of 𝐸_𝑘.
Let 𝑃_𝑘(𝑧)=1𝐸_𝑘𝐸𝑘𝑀(𝑢)𝜔𝑧₀,𝑢𝑑𝑢𝜔𝑧₀,𝑧𝜒_𝑘(𝑧). (22) Then _ℂ𝑛𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)𝜔𝑧₀=,𝑧𝑑𝑧_𝐸𝑘𝑀(𝑧)𝜔𝑧₀,𝑧𝑑𝑧−_𝐸𝑘𝑀(𝑧)𝜔𝑧₀,𝑧𝑑𝑧=0. (23) Without loss of generality, we can assume that 𝒩(𝑀)=1. Then ⋅−𝑧₀^2𝑛𝑏𝑀(⋅)₂^{1−(𝑎/𝑏)}=‖𝑀‖₂^−𝑎/𝑏. (24) Therefore, ⋅−𝑧₀^{2𝑛((1/2)+𝜖)}𝑀(⋅)₂=‖𝑀‖₂^{−𝑎/(𝑏−𝑎)}=𝜎^2𝑛𝑎. (25) Let 𝐵_𝑘={𝑧∈ℂ^𝑛∶|𝑧−𝑧₀|≤2^𝑘𝜎}. Then supp𝑀_𝑘−𝑃_𝑘⊆𝐸_𝑘⊆𝐵_𝑘. (26) In the following, we will prove 1𝐵_𝑘𝐵𝑘𝑀_𝑘−𝑃_𝑘²𝐶𝑑𝑧≤𝐸_𝑘𝐸𝑘𝑀_𝑘(𝑧)²𝑑𝑧. (27) In fact, by _𝐸𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)𝑃_𝑘(𝑧)𝑑𝑧=0, (28) we have 1𝐵_𝑘𝐵𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)²=1𝑑𝑧𝐵_𝑘𝐵𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)=1𝑑𝑧𝐵_𝑘𝐵𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)𝑀_𝑘(𝑧)𝑑𝑧. (29) Since _𝐸𝑘𝑀_𝑘(𝑧)𝑃_𝑘(𝑧)=𝑑𝑧_𝐸𝑘𝑀_𝑘(𝑧)1𝐸_𝑘𝐸𝑘𝑀(𝑢)𝜔𝑧₀,𝑢𝑑𝑢𝜔𝑧₀=1,𝑧𝑑𝑧𝐸_𝑘𝐸𝑘𝑀(𝑢)𝜔𝑧₀,𝑢𝑑𝑢_𝐸𝑘𝑀_𝑘(𝑧)𝜔𝑧₀=𝐸,𝑧𝑑𝑧_𝑘𝑃_𝑘(𝑧)², (30) we get 1𝐵_𝑘𝐵𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)²=1𝑑𝑧𝐵_𝑘𝐸𝑘𝑀_𝑘(𝑧)²𝑑𝑧−_𝐸𝑘𝐸_𝑘𝐵_𝑘𝑃_𝑘(𝑧)²≤1𝑑𝑧𝐵_𝑘𝐸𝑘𝑀_𝑘(𝑧)²𝐶𝑑𝑧≤𝐸_𝑘𝐸𝑘𝑀_𝑘(𝑧)²𝑑𝑧. (31) Therefore, (27) holds true. In particular, we have 1𝐵_0𝐵0𝑀₀(𝑧)−𝑃₀(𝑧)²≤𝐶𝑑𝑧𝐸_0𝐸0𝑀₀(𝑧)²𝑑𝑧≤𝐶𝜎^−2𝑛𝜎^{4𝑛((1/2)−(1/𝑝))}=𝐶𝐵₀^−2/𝑝. (32) For 𝑘≥1, 1𝐵_𝑘𝐵𝑘𝑀_𝑘(𝑧)−𝑃_𝑘(𝑧)²≤𝐶𝑑𝑧𝐸_𝑘𝐸𝑘𝑀_𝑘(𝑧)²=𝐶𝑑𝑧𝐸_𝑘𝐸𝑘𝑀_𝑘(𝑧)²𝑧−𝑧₀^{2𝑛(1+2𝜖)}×𝑧−𝑧₀^{−2𝑛(1+2𝜖)}2𝑑𝑧≤𝐶^𝑘𝜎^−2𝑛2^𝑘−1𝜎^{−2𝑛(1+2𝜖)}×_𝐸𝑘𝑀_𝑘(𝑧)²𝑧−𝑧₀^{2𝑛(1+2𝜖)}𝑑𝑧≤𝐶𝜎^{−4𝑛(1+𝜖)}2^{−4𝑘𝑛−4𝑘𝑛𝜖}𝜎^4𝑛𝑎≤𝐶2^{−4𝑘𝑛−4𝑘𝑛𝜖−4𝑘𝑛/𝑝}2^𝑘𝜎^{−2𝑛/𝑝}=𝐶2^{−4𝑘𝑛𝑎}𝐵_𝑘^−2/𝑝, (33) where 𝐶 depends on 𝑛,𝜖. This proves that 𝑀_𝑘−𝑃_𝑘=𝜆_𝑘𝑎_𝑘, where 𝑎_𝑘 is a 𝐻_𝐿^𝑝,2-atom supported on 𝐵_𝑘 and |𝜆_𝑘|≤𝐶2^{−2𝑘𝑛𝑎}.
Now, we prove that ∑^∞_𝑘=1𝑃_𝑘(𝑧) has atomic decomposition. For 𝑘≥1, 𝑃_𝑘(𝑧≤1)𝐸_𝑘𝐸𝑘|𝑀(𝑢=1)|𝑑𝑢𝐸_𝑘𝐸𝑘𝑢−𝑧₀^2𝑛𝑏|𝑀(𝑢)|𝑢−𝑧₀^−2𝑛𝑏𝑑𝑢≤𝐶(2^𝑘𝜎)^−2𝑛𝑏1𝐸_𝑘⋅−𝑧₀^2𝑛𝑏𝑀(⋅)₂𝐸_𝑘^1/2≤𝐶2^𝑘𝜎^{−2𝑛𝑏−𝑛}𝜎^2𝑛𝑎=𝐶2^{−𝑘𝑛(1+2𝑏)}𝜎^{−2𝑛/𝑝}. (34) Therefore, 𝑃_𝑘(𝑧)=^∞_𝑙=𝑘+1𝑃_𝑙−1(𝑧)−𝑃_𝑙(𝑧). (35) Let 𝑁^𝑘=^∞_{𝑙=𝑘𝐸𝑙}𝑀(𝑢)𝜔𝑧₀,𝑢𝑑𝑢. (36) Then, 𝑁⁰=^∞_{𝑙=0𝐸𝑙}𝑀(𝑢)𝜔𝑧₀=,𝑢𝑑𝑢_ℂ𝑛𝑀(𝑢)𝜔𝑧₀,𝑢𝑑𝑢=0. (37) Thus, by Abel transform, ^∞_𝑘=0𝑃_𝑘(𝑧)=^∞∞_{𝑘=0𝑙=𝑘+1}𝑃_𝑙−1(𝑧)−𝑃_𝑙(𝑧=_∞𝑘=0)𝑁^𝑘+1𝐸_𝑘⁻¹𝜔𝑧₀,𝑧𝜒_𝑘(𝑧)−𝐸_𝑘+1⁻¹𝜔𝑧₀,𝑧𝜒_𝑘+1(𝑧). (38) Following from (34), we obtain 𝑁^𝑘+1𝐸_𝑘⁻¹𝜔𝑧₀,𝑧𝜒_𝑘(𝑧)−𝐸_𝑘+1⁻¹𝜔𝑧₀,𝑧𝜒_𝑘+1(𝑧)≤𝐶2^{−2𝑛(𝑘+1)𝜖}𝜎^{2𝑛−(2𝑛/𝑝)}𝐵_𝑘+1⁻¹=𝐶2^{−2𝑛(𝑘+1)𝜖}𝜎^{2𝑛−(2𝑛/𝑝)}2^𝑘+1𝜎⁻¹=𝐶2^{−2𝑛𝑎(𝑘+1)}𝐵_𝑘+1^−1/𝑝. (39) Let 𝜇_𝑘=𝐶2^{−2𝑛𝑎(𝑘+1)} and 𝑏_𝑘(𝑧)=𝐶⁻¹2^{2𝑛𝑎(𝑘+1)}𝑁^𝑘+1𝐸_𝑘⁻¹𝜔𝑧₀,𝑧𝜒_𝑘(𝑧)−𝐸_𝑘+1⁻¹𝜔𝑧₀,𝑧𝜒_𝑘+1(𝑧). (40) Then, 𝑏_𝑘 are 𝐻_𝐿^𝑝,∞-atoms, ∑^∞_𝑘=0|𝜇_𝑘|^𝑝<∞, and ^∞_𝑘=0𝑃_𝑘(𝑧)=^∞_𝑘=0𝜇_𝑘𝑏_𝑘(𝑧). (41) Therefore, 𝑀(𝑧)=^∞_𝑘=0𝜆_𝑘𝑎_𝑘(𝑧)+^∞_𝑘=0𝜇_𝑘𝑏_𝑘(𝑧) (42) holds pointwise, where 𝑎_𝑘 are 𝐻_𝐿^𝑝,2-atoms and 𝑏_𝑘 are 𝐻_𝐿^𝑝,∞-atoms, and ^∞_𝑘=0𝜆_𝑘^𝑝+𝜇_𝑘^𝑝<∞. (43) When 𝑝=1, it is easy to see that the sum in (42) converges in 𝐿¹.
To prove 𝑀∈𝐻_𝐿^𝑝,𝑞 for 2𝑛/(2𝑛+1)<𝑝<1, we need to show that, for every 𝑔∈Λ^𝐿_(1/𝑝)−1, _ℂ𝑛𝑀(𝑧)𝑔(𝑧)𝑑𝑧=lim^𝑚_{𝑚→∞𝑘=0ℂ𝑛}𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧)𝑔(𝑧)𝑑𝑧. (44) In fact, (44) implies that (42) holds in 𝒮^′(ℂ^𝑛).
For any 𝑧∈ℂ^𝑛, there exists 𝑘≥0 such that 𝑧∈𝐸_𝑘. If 𝑘=0, then 𝑀(𝑧)=𝜆₀𝑎₀(𝑧)+𝜇₀𝑏₀(𝑧). (45) If 𝑘≥1, then 𝑀(𝑧)=𝜆_𝑘(𝑧)𝑎_𝑘(𝑧)+^𝑘_{𝑗=𝑘−1}𝜇_𝑗𝑏_𝑗(𝑧). (46) Therefore, when |𝑧−𝑧₀|≤2^𝑚𝜎, ^∞_𝑘=0𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧=_𝑚𝑘=0)𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧)=𝑀(𝑧). (47) Thus, _{{𝑧∶|𝑧−𝑧0|≤2𝑚}^𝑚_𝜎}𝑘=0𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧𝑔(𝑧)=)𝑑𝑧_{{𝑧∶|𝑧−𝑧0|≤2𝑚𝜎}}𝑀(𝑧)𝑔(𝑧)𝑑𝑧. (48) Let 𝑚→∞; the right side is ∫_ℂ𝑛𝑀(𝑧)𝑔(𝑧)𝑑𝑧. The left side is lim^𝑚_{𝑚→∞𝑘=0{𝑧∶|𝑧−𝑧0|≤2𝑚𝜎}}𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧)𝑔(𝑧)𝑑𝑧=lim^𝑚_{𝑚→∞𝑘=0ℂ𝑛}𝜆_𝑘𝑎_𝑘(𝑧)+𝜇_𝑘𝑏_𝑘(𝑧)𝑔(𝑧)𝑑𝑧. (49) 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 Theorem 8. Theorem 8 follows from Lemmas 11 and 12.

3. The Boundedness of Local Riesz Transform on 𝐻^𝑝_𝐿(ℂ^𝑛)

In this section, we prove the boundedness of local Riesz transform on 𝐻^𝑝_𝐿(ℂ^𝑛) by using Theorem 8.

Proof of Theorem 9. By Theorem 8, it is sufficient to prove that, for any 𝐻_𝐿^𝑝,2-atom 𝑎, 𝑅_𝑗(𝑎) is a 𝐻_𝐿^𝑝,2,𝜀-molecule and the norm 𝒩(𝑅_𝑗(𝑎))≤𝐶, where 𝐶 is independent of 𝑎.
Assume that supp𝑎⊂𝐵(𝑧₀,𝑟); then _ℂ𝑛𝑅_𝑗(𝑎)(𝑧)𝜔𝑧₀=,𝑧𝑑𝑧_{ℂ𝑛ℂ𝑛}𝑎()𝑢𝑧−𝑢^𝑖|𝑢|^2𝑛+1𝜓(𝑢)𝜔()×𝑧,𝑢𝑑𝑢𝜔𝑧₀=,𝑧𝑑𝑧_ℂ𝑛𝑢_𝑖|𝑢|^2𝑛+1𝜓(𝑢)_ℂ𝑛𝑎(𝑧−𝑢)𝜔𝑢+𝑧₀×,𝑧𝑑𝑧𝜔𝑧₀,𝑧𝑑𝑢=0, (50) where the last equality is valid because 𝑎(⋅−𝑢) is an atom supported on 𝐵(𝑢+𝑧₀,𝑟). This proves that 𝑅_𝑗(𝑎) satisfies moment condition.
Denote 𝑀(𝑧)=𝑅_𝑗(𝑎)(𝑧). Then, we have ‖𝑀‖₂=𝑅_𝑗(𝑎)₂≤𝐶‖𝑎‖₂≤|𝐵|^1/2−1/𝑝=𝐶|𝐵|^𝑎−𝑏. (51) Let 𝐵^∗={𝑧∈ℂ^𝑛∶|𝑧−𝑧₀|≤2𝑟}. Then, _ℂ𝑛𝑧−𝑧₀^{2𝑛(1+2𝜖)}|𝑀(𝑧)|²=𝑑𝑧_𝐵∗𝑧−𝑧₀^{2𝑛(1+2𝜀)}𝑀(𝑧)²+𝑑𝑧_{(𝐵∗)𝑐}𝑧−𝑧₀^{2𝑛(1+2𝜖)}|𝑀(𝑧)|²𝑑𝑧=𝐼+𝐼𝐼. (52) For 𝐼, 𝐼≤𝐶|𝐵|^1+2𝜖_ℂ𝑛|𝑀(𝑧)|²𝑑𝑧≤𝐶|𝐵|^{2+2𝜖−(2/𝑝)}=𝐶|𝐵|^2𝑎. (53) For 𝐼𝐼, since 𝑅_𝑗(𝑎𝑧=)()_ℂ𝑛𝑎(𝑢)𝑧_𝑗−𝑢_𝑖|𝑧−𝑢|^2𝑛+1𝜓(𝑧−𝑢)𝜔()=𝑧,𝑢𝑑𝑢_ℂ𝑛𝑎(𝑢)𝜔𝑧₀𝑧,𝑢^𝑗−𝑢_𝑖|𝑧−𝑢|^2𝑛+1𝜓()𝑧𝑧−𝑢×𝜔₀−𝑧−𝑧,𝑢_𝑗−𝑧_0,𝑗𝑧−𝑧₀^2𝑛+1𝜓𝑧−𝑧₀×𝜔𝑧₀−𝑧,𝑧₀𝑑𝑢≤𝐶_ℂ𝑛𝑢−𝑧₀𝑧−𝑧₀^2𝑛+1|𝑎(𝑢|𝐵|)|𝑑𝑢≤𝐶𝑟^1−1/𝑝1𝑧−𝑧₀^2𝑛+1, (54) we get 𝐼𝐼≤𝐶𝑟²|𝐵|^2−2/𝑝_{(𝐵∗)𝑐}𝑧−𝑧₀^{2𝑛(1+2𝜖)}𝑧−𝑧₀^4𝑛+2𝑑𝑧. (55) Let 0<𝜖<1/2𝑛. Then 𝐼𝐼≤𝐶|𝐵|^{2+2𝜖−2/𝑝}=𝐶|𝐵|^2𝑎. (56) Therefore, 𝒩(𝑀)=‖𝑀‖₂^𝑎/𝑏⋅−𝑧₀^{𝑛(1+2𝜖)}𝑀(⋅)₂^{1−(𝑎/𝑏)}≤𝐶|𝐵|^{(𝑎/𝑏)(𝑎−𝑏)}|𝐵|^{𝑎(1−(𝑎/𝑏))}=𝐶. (57) This completes the proof of Theorem 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11001002 and 10901018), the Beijing Natural Science Foundation (1142005), and the Program for New Century Excellent Talents in University.