Characterizations of Multiparameter Besov and Triebel-Lizorkin Spaces Associated with Flag Singular Integrals

Abstract

We introduce the inhomogeneous multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals via the Littlewood-Paley-Stein theory. We establish difference characterizations and Peetre's maximal function characterizations of these spaces.

1. Introduction and Main Results

The flag singular integral operators were first introduced by Müller, Ricci, and Stein when they studied the Marcinkiewicz multiplier on the Heisenberg groups in [1]. To study the □_𝑏-complex on certain CR submanifolds of ℂ^𝑛, in 2001, Nagel et al. [2] studied a class of product singular integrals with flag kernel. They proved, among other things, the 𝐿^𝑝 boundedness of flag singular integrals. More recently, Nagel et al. in [3, 4] have generalized these results to a more general setting, namely, homogeneous group. For other related results, see [5, 6].

For 0<𝑝≤1, Han and Lu [7] developed Hardy spaces 𝐻^𝑝𝔉(ℝ^𝑛×ℝ^𝑚) with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calderón’s identity and proved the 𝐻^𝑝𝔉(ℝ^𝑛×ℝ^𝑚)→𝐻^𝑝𝔉(ℝ^𝑛×ℝ^𝑚) and 𝐻^𝑝𝔉(ℝ^𝑛×ℝ^𝑚)→𝐿^𝑝(ℝ^𝑛×ℝ^𝑚) boundedness for flag singular integral operators. The duality of 𝐻^𝑝𝔉(ℝ^𝑛×ℝ^𝑚) was also established. More recently, Ding et al. studied the homogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals in [8] and proved the boundedness of flag singular integrals on these spaces. Similar results can also be found in [9].

The aim of this paper is to give the new difference characterization as well as Petree’s maximal function characterization of multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals, which reflect that the Besov spaces and Triebel-Lizorkin spaces have flag multiparameter structure. These characterizations are established for the inhomogeneous Besov and Triebel-Lizorkin spaces, but the argument goes through with only minor alterations in the homogeneous ones introduced in [8].

In order to describe more precisely questions and results studied in this paper, we begin with basic notations and notions. Let 𝜓⁽¹⁾∈𝒮(ℝ^𝑛×ℝ^𝑚) withsupp𝜓⁽¹⁾⊂(𝜉,𝜂)∈ℝ^𝑛×ℝ^𝑚∶12≤||𝜉,𝜂||≤2_𝑗∈ℤ,|||𝜓₍₁₎2^−𝑗𝜉,2^−𝑗𝜂|||²=1∀(𝜉,𝜂)∈ℝ^𝑛×ℝ^𝑚⧵{(0,0)},(1.1)

and let 𝜓⁽²⁾∈𝒮(ℝ^𝑚) withsupp𝜓⁽²⁾⊂𝜂∈ℝ^𝑚∶12≤||𝜂||≤2_𝑘∈ℤ,|||𝜓₍₂₎2^−𝑘𝜂|||²=1∀𝜂∈ℝ^𝑚⧵{0}.(1.2)

Let 𝜓^𝑗(1)(𝑥,𝑦)=2^𝑗𝑛𝜓⁽¹⁾(2^𝑗𝑥,2^𝑗𝑦),𝜓^𝑘(2)(𝑦)=2^𝑘𝑚𝜓⁽²⁾(2^𝑘𝑦), and𝜓_𝑗,𝑘(𝑥,𝑦)=𝜓^𝑗(1)∗₂𝜓^𝑘(2)(𝑥,𝑦)=𝜓^𝑗(1)(𝑥,𝑦−𝑧)𝜓^𝑘(2)(𝑧)𝑑z,(1.3)

then, for 𝑓∈𝐿²(ℝ^𝑛×ℝ^𝑚), the following Calderón’s reproducing formula holds:𝑓=_{𝑗∈ℤ𝑘}_∈ℤ𝜓_𝑗,𝑘∗𝜓_𝑗,𝑘∗𝑓,(1.4)

where the series converges in 𝐿²(ℝ^𝑛×ℝ^𝑚).

A Schwartz function 𝑓∈𝒮(ℝ^𝑛+2𝑚) is said to be a product test function in 𝒮_∞(ℝ^𝑛+𝑚×ℝ^𝑚) if it satisfies_{ℝ𝑚×ℝ𝑛}𝑓(𝑥,𝑦,𝑧)𝑥^𝛼𝑦^𝛽𝑑𝑥𝑑𝑦=_ℝ𝑚𝑓(𝑥,𝑦,𝑧)𝑧^𝛾𝑑𝑧=0,(1.5)

for all multi-indices 𝛼∈ℕ^𝑛,𝛽,𝛾∈ℕ^𝑚.

Definition 1.1. A function 𝑓(𝑥,𝑦) defined on ℝ^𝑛×ℝ^𝑚 is said to be a test function in 𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) if there exists a function 𝑓^#∈𝒮_∞(ℝ^𝑚+𝑛×ℝ^𝑚) such that 𝑓(𝑥,𝑦)=_ℝ𝑚𝑓^#(𝑥,𝑦−𝑧,𝑧)𝑑𝑧,(1.6)and the seminorm of 𝑓 is defined by ‖𝑓‖_{𝔉,𝛼,𝛽}=inf‖‖𝑓^#‖‖_𝛼,𝛽∶forallrepresentationsof𝑓in(1.7),(1.7) where ‖⋅‖_𝛼,𝛽 denote the seminorm in 𝒮(ℝ^𝑛+2𝑚). Denote by 𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) the dual of 𝒮_𝔉(ℝ^𝑛×ℝ^𝑚).

Choose a Schwartz function 𝜑 on ℝ^𝑛×ℝ^𝑚 such that||𝜑(𝜉,𝜂)||²=1−^∞_∞𝑗=1_𝑘=1|||𝜓₍₁₎2^−𝑗𝜉₁,2^−𝑗𝜉₂|||²|||𝜓₍₂₎2^−𝑘𝜉₂|||².(1.8)Note that 𝜑∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) with Fourier transform is supported in {(𝜉,𝜂)∶|(𝜉,𝜂)|≤1}. Define the operator 𝑆₀ by 𝑆₀𝑓=𝜑∗𝑓,𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚).

For 𝑓∈𝐿²(ℝ^𝑛×ℝ^𝑚), by taking the Fourier transform,𝑓=𝜑∗𝜑∗𝑓+^∞_∞𝑗=1_𝑘=1𝜓_𝑗,𝑘∗𝜓_𝑗,𝑘∗𝑓,(1.9)

where the series converges in 𝐿²(ℝ^𝑛×ℝ^𝑚) norm.

Now, we introduce the definition of inhomogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals.

Definition 1.2. Let 𝛼,𝛽∈(−∞,∞) and 𝑝,𝑞∈(1,∞). The inhomogeneous Triebel-Lizorkin space associated with flag singular integrals ℱ^{𝛼,𝛽𝑝,𝑞}(ℝ^𝑛×ℝ^𝑚) is defined to be the collection of all 𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) such that ‖𝑓‖^𝜓ℱ_{𝛼,𝛽𝑝,𝑞(ℝ𝑛×ℝ𝑚)}∶=‖‖𝑆₀𝑓‖‖_{𝐿𝑝(ℝ𝑛×ℝ𝑚)}+‖‖‖‖^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞||𝜓_𝑗,𝑘∗𝑓||^𝑞^1/𝑞‖‖‖‖_{𝐿𝑝(ℝ𝑛×ℝ𝑚)}<∞,(1.10) and the inhomogeneous Besov space associated with flag singular integrals ℬ^{𝛼,𝛽𝑝,𝑞}(ℝ^𝑛×ℝ^𝑚) is defined to be the collection of all 𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) such that ‖𝑓‖^𝜓ℬ_{𝛼,𝛽𝑝,𝑞(ℝ𝑛×ℝ𝑚)}∶=‖‖𝑆₀𝑓‖‖_{𝐿𝑝(ℝ𝑛×ℝ𝑚)}+^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝐿_{𝑝(ℝ𝑛×ℝ𝑚)}^1/𝑞<∞.(1.11)

Throughout this paper, we always work on ℝ^𝑛×ℝ^𝑚 for some fixed 𝑛,𝑚 and use ℱ^{𝛼,𝛽𝑝,𝑞} to denote ℱ^{𝛼,𝛽𝑝,𝑞}(ℝ^𝑛×ℝ^𝑚), similarly for 𝒮_𝔉,ℬ^{𝛼,𝛽𝑝,𝑞}, and so forth. We would like to point out that the multiparameter structures are involved in the definitions of ℬ^{𝛼,𝛽𝑝,𝑞} and ℱ^{𝛼,𝛽𝑝,𝑞}. The following result shows that the definition of the Besov spaces ℬ^{𝛼,𝛽𝑝,𝑞} and Triebel-Lizorkin spaces ℱ^{𝛼,𝛽𝑝,𝑞} is independent of the choice of (𝜓⁽¹⁾,𝜓⁽²⁾,𝜑); thus, the Besov spaces ℬ^{𝛼,𝛽𝑝,𝑞} and the Triebel-Lizorkin spaces ℱ^{𝛼,𝛽𝑝,𝑞} are well defined.

Theorem 1.3. If 𝜃_𝑗,𝑘 satisfies the same conditions as 𝜓_𝑗,𝑘, and 𝜙 is defined similar to (1.8) with 𝜓 replaced by 𝜃, then for 𝛼,𝛽∈ℝ and 𝑝,𝑞∈(1,∞) and 𝑓∈𝒮_𝔉, ‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞}∼‖𝑓‖^𝜓ℱ_{𝛼,𝛽𝑝,𝑞},‖𝑓‖^𝜃ℬ_{𝛼,𝛽𝑝,𝑞}∼‖𝑓‖^𝜓ℬ_{𝛼,𝛽𝑝,𝑞}.(1.12)

Remark 1.4. As the classical case, it is not hard to show that ‖⋅‖^𝜓ℬ_{𝛼,𝛽𝑝,𝑞} and ‖⋅‖^𝜓ℱ_{𝛼,𝛽𝑝,𝑞} are norms of ℬ^{𝛼,𝛽𝑝,𝑞} and ℱ^{𝛼,𝛽𝑝,𝑞}, respectively. Moreover, ℬ^{𝛼,𝛽𝑝,𝑞} and ℱ^{𝛼,𝛽𝑝,𝑞} are complete with respect to these norms and hence are Banach spaces. We omit the details.

Throughout this paper, we use the notations 𝑗∧𝑘=min{𝑗,𝑘} and 𝑗∨𝑘=max{𝑗,𝑘}. We introduce the following flag multi-parameter Peetre maximal functions (with respect to 𝜓). For 𝑗,𝑘∈ℤ₊ and 𝑏⃗=(𝑏₁,𝑏₂)∈ℝ₊×ℝ₊, define𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦)=sup_{(𝑢,𝑣)∈ℝ𝑛×ℝ𝑚}||𝜓_𝑗,𝑘∗𝑓(𝑥−𝑢,𝑦−𝑣)||(1+2^𝑗|𝑢|)^𝑏1(1+2^𝑗∧𝑘|𝑣|)^𝑏2.(1.13)

For 𝑗=𝑘=0, define𝜑^{∗⃗𝑏,0,0}(𝑓)(𝑥)=sup_{𝑦∈ℝ𝑛}||𝜑∗𝑓(𝑥−𝑢,𝑦−𝑣)||(1+|𝑢|)^𝑏1(1+|𝑣|)^𝑏2.(1.14)

An index 𝑏⃗=(𝑏₁,𝑏₂) is said to be admissible if𝑏₁>𝑛𝑝∧𝑞,𝑏²>𝑚𝑝∧𝑞.(1.15)

We point out again that the flag multiparameter structure is involved in the definition of Peetre’s maximal functions. The maximal function characterizations of Besov spaces and Triebel-Lizorkin spaces are as follows.

Theorem 1.5. Let 𝛼,𝛽∈ℝ and 𝑝,𝑞∈(1,∞). If 𝑏⃗ is admissible, then for 𝑓∈𝒮_𝔉, one has(i)‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞,(1)}∶=‖𝑆₀𝑓‖_𝑝+‖{2^𝛼𝑗2^𝛽𝑘‖𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)‖_𝑝}_𝑗,𝑘‖_ℓ𝑞∼‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞}, (ii)‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞,(1)}∶=‖𝑆₀𝑓‖_𝑝+‖‖{2^𝛼𝑗2^𝛽𝑘𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)}_𝑗,𝑘‖_ℓ^𝑞‖_𝐿^𝑝∼‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}. Here and in what follows, one uses the following notation: ‖‖{𝑎_𝑗,𝑘}_𝑗,𝑘‖‖_ℓ𝑞∶=^∞_∞𝑗=1_𝑘=1|𝑎_𝑗,𝑘|^𝑞^1/𝑞.(1.16)

In order to state our result for flag singular integrals, we need to recall some definitions given in [2]. Following closely from [2], we begin with the definitions of a class of distributions on an Euclidean space ℝ^𝑑. A 𝑘-normalized bump function on a space ℝ^𝑑 is a 𝐶^𝑘 function supported on the unit ball with 𝐶^𝑘 norm bounded by 1. As pointed out in [2], the definitions given below are independent of the choices of 𝑘, and thus we will simply refer to “normalized bump function’’ without specifying 𝑘.

Definition 1.6. A flag kernel on ℝ^𝑛×ℝ^𝑚 is a distribution 𝐾 on ℝ^𝑛+𝑚 which coincides with a 𝐶^∞ function away from the coordinate subspaces (0,𝑦), where (0,𝑦)∈ℝ^𝑛×ℝ^𝑚 and satisfies(1) (Differential inequalities) for any multi-indices 𝛼 and  𝛽, ||𝜕^𝛼𝑥𝜕^𝛽𝑦𝐾(𝑥,𝑦)||≤𝐶_𝛼,𝛽|𝑥|^{−𝑛−|𝛼|}(|𝑥|+|𝑦|)^{−𝑚−|𝛽|},(1.17) for all (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑚 with |𝑥|≠0,(2) (Cancellation condition) ||||_ℝ𝑚𝜕^𝛼𝑥𝐾(𝑥,𝑦)𝜙₁(𝛿𝑦)𝑑𝑦||||≤𝐶_𝛼|𝑥|^{−𝑛−|𝛼|},(1.18) for all multi-index 𝛼 and every normalized bump function 𝜙₁ on ℝ^𝑚 and every 𝛿>0, ||||_ℝ𝑛𝜕^𝛽𝑦𝐾(𝑥,𝑦)𝜙₂(𝛿𝑥)𝑑𝑥||||≤𝐶_𝛽|𝑦|^{−𝑚−|𝛽|}(1.19) for every multi-index 𝛽 and every normalized bump function 𝜙₂ on ℝ^𝑛 and every 𝛿>0, ||||_ℝ𝑛+𝑚𝐾(𝑥,𝑦)𝜙₃𝛿₁𝑥,𝛿₂𝑦𝑑𝑥𝑑𝑦||||≤𝐶,(1.20) for every normalized bump function 𝜙₃ on ℝ^+𝑚 and every 𝛿₁>0 and 𝛿₂>0.

The boundedness of flag singular integrals on these inhomogeneous Besov spaces and Triebel-Lizorkin spaces is given by the following theorem, whose proof is quite similar to that in homogeneous case in [8]. We omit the proof here.

Theorem 1.7. Suppose that 𝑇 is a flag singular integral defined on ℝ^𝑛×ℝ^𝑚 with the flag kernel 𝐾, then, for 𝑝,𝑞>1 and 𝛼,𝛽∈ℝ, 𝑇 is bounded on ℱ^{𝛼,𝛽𝑝,𝑞} and on ℬ^{𝛼,𝛽𝑝,𝑞}.

As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will give the difference characterization for ℬ^{𝛼,𝛽𝑝,𝑞} and ℱ^{𝛼,𝛽𝑝,𝑞}. However, the new feature is that the differences of functions are associated with the “flag.’’ More precisely, for (𝑢,𝑣)∈ℝ^𝑛×ℝ^𝑚 and 𝑤∈ℝ^𝑚, we define the first flag difference (associated the flag {(0,0)}⊂{(0,𝑦)}) in ℝ^𝑛×ℝ^𝑚 byΔ^{𝔉𝑢,𝑣;𝑤}𝑓(𝑥,𝑦)==𝑓Δ(𝑥^𝑢,𝑣+∘𝑢Δ,𝑦^𝑤(2)+𝑓𝑣(+𝑥,𝑤𝑦))−𝑓(𝑥+𝑢,𝑦+𝑣)−𝑓(𝑥,𝑦+𝑤)+𝑓(𝑥,𝑦),(1.21)

where Δ_𝑢,𝑣 is the difference operator on ℝ^𝑚+𝑛, and Δ^𝑤(2) is the difference operator on ℝ^𝑚. For 𝑘∈ℤ₊ and 𝑘≥2, the 𝑘th flag difference operator (Δ^{𝔉𝑢,𝑣;𝑤})^𝑘 can be defined inductively by(Δ^{𝔉𝑢,𝑣;𝑤})^𝑘=Δ^{𝔉𝑢,𝑣;𝑤}∘(Δ^{𝔉𝑢,𝑣;𝑤})^𝑘−1.(1.22)

Theorem 1.8. If 𝛼,𝛽>0, 1<𝑝<∞, 1<𝑞<∞, and 𝑀≥[𝛼∨𝛽]+1, where [⋅] denotes the greatest integer function, one defines ‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞,(2)}∶=‖𝑓‖_𝑝+⎧⎪⎨⎪⎩_{ℝ𝑛+𝑚×ℝ𝑚}⎛⎜⎜⎝‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖_𝑝|(𝑢,𝑣)|^𝛼|𝑤|^𝛽⎞⎟⎟⎠^𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|^𝑛+𝑚|𝑤|^𝑚⎫⎪⎬⎪⎭^1/𝑞‖𝑓‖_{ℱ𝛼,𝛽𝑝,𝑞,(2)},∶=‖𝑓‖_𝑝+‖‖‖‖∫_{ℝ𝑛+𝑚×ℝ𝑚}|(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓||(𝑢,𝑣)|^𝛼|𝑤|^𝛽^𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|^𝑛+𝑚|𝑤|^𝑚^1/𝑞‖‖‖‖_𝑝,(1.23) then ‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞,(2)}∼‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞},‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞,(2)}∼‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}.

As mentioned before, by slightly modifying the proof, we can prove difference characterizations and Peetre’s maximal function characterizations of homogeneous Besov and Triebel-Lizorkin spaces, introduced in [8]. We leave the details to the interested reader.

The following of the paper is organized as follows. In Section 2, we give some lemmas. The proof of Theorems 1.3 and 1.5 is presented in Section 3. Section 4 is devoted to the proof of Theorem 1.8.

2. Some Lemmas

In this section, we present some lemmas, which will be used in the proofs of the theorems.

2.1. Inhomogeneous Calderón’s Reproducing Formula in 𝒮^𝔉

Lemma 2.1. The inhomogeneous Calderón’s reproducing formula holds 𝑓=𝜑∗𝜑∗𝑓+^∞_∞𝑗=1_𝑘=1𝜓_𝑗,𝑘∗𝜓_𝑗,𝑘∗𝑓,(2.1) where the series converges in 𝐿^𝑝(ℝ^𝑛+𝑚)(1<𝑝<∞), 𝒮_𝔉(ℝ^𝑛×ℝ^𝑚) and 𝒮^𝔉(ℝ^𝑛×ℝ^𝑚).

We point out that in [8] the homogeneous Calderón’s reproducing formula was provided. Note that the convergence of these two kind of producing formulas are different. See [8] for homogeneous case.

Proof. For any 𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚), then by definition, there exists 𝑓^#∈𝒮(ℝ^𝑛+𝑚×ℝ^𝑚) such that 𝑓(𝑥,𝑦)=∫_ℝ𝑚𝑓^#(𝑥,𝑦−𝑧,𝑧)𝑑𝑧. We need to show that for all 𝑁∈ℤ₊, 𝜑^#∗𝜑^#∗𝑓^#+_{𝑗,𝑘∈ℤ+∶𝑗≤𝑁or𝑘≤𝑁}𝜓^#𝑗,𝑘∗𝜓^#𝑗,𝑘∗𝑓^#,(2.2) tends to 𝑓 in the topology of 𝒮(ℝ^𝑛+2𝑚) as 𝑁→+∞. We only consider the case when 𝑘<𝑁 in the summation in (2.2) since the other case can be dealt with in the same way. Denote, in this case, the expression (2.2) by 𝑓^#𝑁. By Fourier inversion, 𝑓^#(𝑥,𝑦)−𝑓^#𝑁(𝑥,𝑦)=𝑓^#−𝑓^#𝑁^∨(𝑥,𝑦),forevery(𝑥,𝑦)∈ℝ^𝑛+𝑚×ℝ^𝑚.(2.3) Let ℎ_𝑁(𝜉,𝜂,𝑥,𝑦)=𝑒^{𝑖(𝑥⋅𝜉+𝑦⋅𝜂)}𝑓^#(𝜉,𝜂)∑^{∞𝑘=𝑁+1}|𝜓⁽²⁾(2^−𝑘𝜂)|² so that 𝑓^#(𝑥,𝑦)−𝑓^#𝑁(𝑥,𝑦)=𝑐_𝑛_{ℝ𝑛+𝑚×ℝ𝑚}ℎ_𝑁(𝜉,𝜂,𝑥,𝑦)𝑑𝜂𝑑𝜉.(2.4) Since |ℎ_𝑁(𝜉,𝜂,𝑥,𝑦)|≤|𝑓^#(𝜉,𝜂)|∈𝐿¹(ℝ^𝑛+𝑚×ℝ^𝑚), Lebesgue’s dominated convergence theorem yields lim_𝑁→+∞𝑓^#𝑁(𝑥,𝑦)=𝑓^#(𝑥,𝑦),forevery(𝑥,𝑦)∈ℝ^𝑛+𝑚×ℝ^𝑚.(2.5)
On the other hand, using the cancellation conditions of 𝜓^𝑗(1)∗𝜓^𝑗(1),𝜓^𝑘(2)∗𝜓^𝑘(2), and the smoothness of 𝑓^#, we can get ||𝜓^#𝑗,𝑘∗𝜓^#𝑗,𝑘∗𝑓^#(𝑥,𝑦)||≲2^−𝑗2^−𝑘1+|𝑥|+||𝑦||^−𝐿,∀𝑗,𝑘,𝐿∈ℤ₊.(2.6)
Now, (2.5) together with the estimate (2.6) implies that, for 𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚), ||𝑓^#(𝑥)−𝑓^#𝑁(𝑥,𝑦)||=_{𝑗∈ℤ+𝑘≥}_𝑁+1||𝜓^#𝑗,𝑘∗𝜓^#𝑗,𝑘∗𝑓^#(𝑥,𝑦)||≲2^−𝑁1+|𝑥|+||𝑦||^−𝐿.(2.7) Applying this to 𝜕^𝑢𝑓^# (here 𝑢 denotes any multi-index in ℕ^𝑛+2𝑚) and noting that 𝜕^𝑢𝑓^#𝑁=(𝜕^𝑢𝑓^#)_𝑁, we obtain lim_𝑁→+∞sup_{(𝑥,𝑦)∈ℝ𝑛×ℝ𝑚}(1+|𝑥|+||𝑦||)^𝐿||𝜕^𝑢𝑓^#−𝑓^#𝑁(𝑥,𝑦)||=0.(2.8) This proves the convergence of series in (2.1) in 𝒮_𝔉(ℝ^𝑛×ℝ^𝑚). The convergence in 𝒮^𝔉(ℝ^𝑛×ℝ^𝑚) follows by a standard duality argument. The convergence in 𝐿^𝑝(ℝ^𝑛+𝑚) can be proved similar to the product case, see (10, Theorem  1.1).

2.2. Almost Orthogonality Estimates

The following lemma is the almost orthogonality estimates, which will be frequently used. See [7] for a proof.

Lemma 2.2. Let 𝑥∈ℝ^𝑛,𝑦∈ℝ^𝑚. Given any positive integers 𝐿 and 𝑀, there exists a constant 𝐶=𝐶(𝐿,𝑀)>0 such that ||𝜓_𝑗,𝑘∗𝜑_𝑗′,𝑘^′(𝑥,𝑦)||≤𝐶2^{−|𝑗−𝑗′|𝐿}2^{−|𝑘−𝑘′|𝐿}2^{(𝑗∧𝑗′)𝑀}(2^{𝑗∧𝑗′}+|𝑥|)^𝑛+𝑀2^{(𝑗∧𝑗′∧𝑘∧𝑘′)𝑀}(2^{𝑗∧𝑗′∧𝑘∧𝑘′}+||𝑦||)^𝑚+𝑀,(2.9) where 𝜓,𝜑 are defined as in Section 1.

2.3. Maximal Function Estimates

The maximal function estimates are given as follows.

Lemma 2.3. For 𝑗,𝑘,𝑗^,𝑘^∈ℤ₊, and for any 𝐿>0 and 𝑏⃗=(𝑏₁,𝑏₂)∈ℝ₊×ℝ₊, there exists a constant 𝐶=𝐶(𝐿,𝑏)⃗ depending only on 𝐿 and 𝑏⃗, but independent on 𝑗,𝑘,𝑗^,𝑘^, such that ||𝜓_𝑗,𝑘∗𝜓_𝑗′,𝑘^′∗𝜓_𝑗′,𝑘^′∗𝑓(𝑥,𝑦)||≤𝐶2^{−𝐿|𝑗−𝑗′|}2^{−𝐿|𝑘−𝑘′|}𝜓^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦).(2.10)

Proof. By the almost orthogonality estimate in Lemma 2.2, for any 𝐿>0 and 𝑀>𝑏₁∨𝑏₂, we have the pointwise estimate ||𝜓≲_𝑗,𝑘∗2𝜓_𝑗′,𝑘^′∗_{−|𝑗−𝑗′|𝐿}𝜓_𝑗′,𝑘^′∗𝑓(𝑥,𝑦)||2^{−|𝑘−𝑘′|𝐿′}×_{ℝ𝑛×ℝ𝑚}2^{(𝑗∧𝑗′)𝑛}2^{(𝑗∧𝑘∧𝑗′∧𝑘′)𝑚}||𝜓_{𝑗′,𝑘′}∗𝑓(𝑥−𝑢,𝑦−𝑣)||1+2^{𝑗∧𝑗′}|𝑢|^𝑀+𝑛1+2^{𝑗∧𝑘∧𝑗′∧𝑘′}|𝑣|^𝑀+𝑚≤2_{−|𝑗−𝑗′|𝐿′}𝑑𝑢𝑑𝑣×2^{−|𝑘−𝑘′|𝐿′}𝜓^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥)_{ℝ𝑛×ℝ𝑚}2^{(𝑗∧𝑗′)𝑛}2^{(𝑗∧𝑘∧𝑗′∧𝑘′)𝑚}(1+2^𝑗′|𝑢|)^𝑏1(1+2^{𝑗′∧𝑘′}|𝑣|)^𝑏21+2^{𝑗∧𝑗′}|𝑢|^𝑀+𝑛(1+2^{𝑗∧𝑘∧𝑗′∧𝑘′}|𝑣|)^𝑀+𝑚≲2_{−|𝑗−𝑗′|(𝐿′−𝑏1)}𝑑𝑢𝑑𝑣×2^{−|𝑘−𝑘′|(𝐿′−𝑏1−𝑏2)}𝜓^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦)_{ℝ𝑛×ℝ𝑚}2^{(𝑗∧𝑗′)𝑛}2^{(𝑗∧𝑘∧𝑗′∧𝑘′)𝑚}1+2^{𝑗∧𝑗′}|𝑢|^{𝑀+𝑛−𝑏1}(1+2^{𝑗∧𝑘∧𝑗′∧𝑘′}|𝑣|)^{𝑀+𝑚−𝑏2}≲2_{−|𝑗−𝑗′|(𝐿)}𝑑𝑢𝑑𝑣2^{−|𝑘−𝑘′|(𝐿)}𝜓^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦),(2.11) where 𝐿=𝐿^−𝑏₁−𝑏₂. This proves (2.10).

Remark 2.4. Since the almost orthogonality estimates hold with 𝜓_0,0 or 𝜃_0,0 is replaced by 𝜑, repeating the same argument as (2.11), we see that the estimate (2.10) is still valid if 𝜓_0,0 or 𝜃_0,0 is replaced by 𝜑.

Denote by ℳ_𝑠 the strong maximal operator defined byℳ_𝑠(𝑓)(𝑥)=sup_𝑅∋𝑥_𝑅||𝑓(𝑦)||𝑑𝑦,(2.12)

where the supremum is taken over all open rectangles 𝑅 in ℝ^𝑛×ℝ^𝑚 that contain the point 𝑥.

Lemma 2.5. Let 0<𝑐₁,𝑐₂<∞, and 0<𝑟<∞, then for all 𝑗,𝑘∈ℤ and for all 𝐶¹ functions 𝑢 on ℝ^𝑛×ℝ^𝑚 whose Fourier transform is supported in the rectangle {𝜉∶|𝜉′|≤𝑐₁2^𝑗,|𝜉_𝑛|≤𝑐₂2^𝑗∧𝑘}, one has the estimate sup_{𝑦∈ℝ𝑛}||𝑢(𝑥−𝑢,𝑦−𝑣)||(1+2^𝑗|𝑢|)^𝑛/𝑟(1+2^𝑗∧𝑘|𝑣|)^𝑚/𝑟≤𝐶ℳ_𝑠(|𝑢|^𝑟)(𝑥,𝑦)^1/𝑟.(2.13) In particular, if 𝑏⃗=(𝑏₁,𝑏₂) with 𝑏₁≥𝑛/𝑟 and 𝑏₂≥𝑚/𝑟, then for all 𝑓∈𝐿^𝑝(ℝ^𝑛×ℝ^𝑚)(1<𝑝<∞), 𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦)≲ℳ_𝑠||𝜓_𝑗,𝑘∗𝑓||^𝑟(𝑥,𝑦)^1/𝑟.(2.14)

Lemma 2.5 can be proved as in the classical one-parameter case. We refer the reader to [11].

2.4. An Embedding Result

The following lemma is an embedding result.

Lemma 2.6. For 𝛼,𝛽>0 and 𝑝,𝑞∈(1,∞), one has the following continuous embedding: ℬ^{𝛼,𝛽𝑝,𝑞}↪𝐿^𝑝,ℱ^{𝛼,𝛽𝑝,𝑞}↪𝐿^𝑝.(2.15)

Proof. For 𝑓∈𝒮_𝔉(ℝ^𝑛×ℝ^𝑚), by inhomogeneous Calderón’s reproducing formula (2.1), ‖𝑓‖_𝑝≲‖‖𝑆₀𝑓‖‖_𝑝+^∞_∞𝑗=1_𝑘=1‖‖𝜓_𝑗,𝑘∗𝑓‖‖_𝑝≲‖‖𝑆₀𝑓‖‖_𝑝+^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝^1/𝑞∼‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞},(2.16) where we have used Hölder’s inequality in the last inequality. This proves Lemma  2.6 for Besov spaces.
For the Triebel-Lizorkin spaces, by (2.1), the pointwise inequality |𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)|≤𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦), Hölder’s inequality, and Fefferman-Stein’s vector-valued inequality, we have ‖𝑓‖_𝑝≲‖‖𝜑∗(𝑆₀𝑓)‖‖_𝑝+‖‖‖‖^∞_∞𝑗=1_𝑘=1|𝜓_𝑗,𝑘∗𝜓_𝑗,𝑘∗𝑓|‖‖‖‖_𝑝≲‖‖𝑆₀𝑓‖‖_𝑝+‖‖‖‖^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)^𝑞^1/𝑞‖‖‖‖_𝑝≲‖‖𝑆₀𝑓‖‖_𝑝+‖‖‖‖^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞|𝜓_𝑗,𝑘∗𝑓|^𝑞^1/𝑞‖‖‖‖_𝑝∼‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}.(2.17) This ends the proof of Lemma 2.6.

3. Proof of Theorems 1.3 and 1.5

We first prove Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces by showing‖𝑓‖^𝜓ℱ_{𝛼,𝛽𝑝,𝑞}≲‖𝑓‖^𝜓ℱ_{𝛼,𝛽𝑝,,𝑞,(1)}≲‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,,𝑞,(1)}≲‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞}.(3.1)

The first inequality in (3.1) follows from the pointwise inequality||𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)||≤𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦),∀𝑗,𝑘∈ℤ₊.(3.2)

Next, for any admissible 𝑏, fix 𝑏⃗. Since 𝑏⃗ is admissible, we can choose 𝑟≤𝑝∧𝑞 such that 𝑏₁≥𝑛/𝑟, 𝑏₂≥𝑚/𝑟, and the thus inequality (2.14) holds. We apply Lemma 2.5 and the 𝐿^𝑝/𝑟(ℓ^𝑞/𝑟) boundedness of ℳ_𝑠 to deduce‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞,(1)}≲‖‖‖2^𝑗𝛼2^𝑘𝛽𝜃^{∗⃗𝑏,𝑗,𝑘}(𝑓)_𝑗,𝑘‖‖‖_{𝐿𝑝(ℓ𝑞)}≲‖‖2^𝑗𝛼2^𝑘𝛽ℳ_𝑠(|𝜃_𝑗,𝑘∗𝑓|^𝑟)^1/𝑟_𝑗,𝑘‖‖_{𝐿𝑝(ℓ𝑞)}≲‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞},(3.3)

which gives the third inequality in (3.1).

Thus, to finish the proof of (3.1), it remains to verify the second inequality. For 𝑗,𝑘∈ℤ₊, 𝜙∗𝜃_𝑗,𝑘 is nonzero only when 𝑗=1 and 𝑘=1. Thus, applying Calderón’s identity (2.1), Minkowski’s inequality, Remark 2.4, and Lemma 2.5, we deduce that‖𝜑∗𝑓‖_𝑝≲‖𝜑∗𝜙‖₁⋅‖𝜙∗𝑓‖_𝑝+‖‖𝜃^{∗⃗𝑏,1,1}(𝑓)‖‖_𝑝≲‖𝜙∗𝑓‖_𝑝+‖‖‖2^𝑗𝛼2^𝑘𝛽𝜃^{∗⃗𝑏,𝑗,𝑘}(𝑓)_𝑗,𝑘‖‖‖_{𝐿𝑝(ℓ𝑞)}≲‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞,(1)}.(3.4)

To finish the proof, it remains to show‖‖‖2_𝑗𝛼2^𝑘𝛽𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)_𝑗,𝑘‖‖‖_{𝐿𝑝(ℓ𝑞)}≲‖𝑓‖^𝜃ℱ_{𝛼,𝛽𝑝,𝑞,(1)}.(3.5)

By the inhomogeneous Calderón’s identity (2.1), we have||𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)||≤||𝜓_𝑗,𝑘∗𝜙||∗||𝜙∗𝑓||(𝑥,𝑦)+_{𝑗′,𝑘′∈ℤ+}||𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′||∗||𝜃_{𝑗′,𝑘′}∗𝑓||(𝑥,𝑦).(3.6)

It follows that‖‖‖2_𝑗𝛼2^𝑘𝛽𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)_𝑗,𝑘‖‖‖_{𝐿𝑝(ℓ𝑞)}≤‖‖‖‖2^𝑗𝛼2^𝑘𝛽sup_{(𝑢,𝑣)∈ℝ𝑛×ℝ𝑚}(|𝜓_𝑗,𝑘∗𝜙|∗|𝜙∗𝑓|)(⋅−𝑢,⋅−𝑣)(1+2^𝑗|𝑢|)^𝑏1(1+2^𝑗∧𝑘|𝑣|)^𝑏2_𝑗,𝑘‖‖‖‖_{𝐿𝑝(ℓ𝑞)}+‖‖‖‖‖⎧⎪⎨⎪⎩2^𝑗𝛼2^𝑘𝛽_𝑗′_,𝑘′>0sup_{(𝑢,𝑣)∈ℝ𝑛×ℝ𝑚}(|𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′|∗|𝜃_𝑗′,𝑘^′∗𝑓|)(⋅−𝑢,⋅−𝑣)(1+2^𝑗|𝑢|)^𝑏1(1+2^𝑗∧𝑘|𝑣|)^𝑏2⎫⎪⎬⎪⎭_𝑗,𝑘‖‖‖‖‖_{𝐿𝑝(ℓ𝑞)}∶=𝐼₁+𝐼₂.(3.7)

We first estimate 𝐼₁. By the support properties of 𝜓_𝑗,𝑘 and 𝜑 and Young’s inequality,𝐼₁≲‖‖𝜓_1,1∗𝜙‖‖₁⋅‖𝜙∗𝑓‖_𝑝≲‖𝜙∗𝑓‖_𝑝.(3.8)

Next, we give the estimate for 𝐼₂. For any (𝑢,𝑣)∈ℝ^𝑛×ℝ^𝑚,||𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′||∗||𝜃_{𝑗′,𝑘′}∗𝑓||(𝑥−𝑢,𝑦−𝑣)1+2^𝑗|𝑢|^𝑏11+2^𝑗∧𝑘|𝑣|^𝑏2≲_{ℝ𝑛×ℝ𝑚}||𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′𝑢^,𝑣^||||𝜃_𝑗^′_,𝑘^′∗𝑓𝑥−𝑢−𝑢^,𝑦−𝑣−𝑣^||1+2^𝑗|𝑢|^𝑏11+2^𝑗∧𝑘|𝑣|^𝑏2𝑑𝑢^𝑑𝑣^≤𝜃×1^{∗⃗𝑏,𝑗}_′,𝑘′+(𝑓2)(_𝑗′𝑥,𝑦)_{ℝ𝑛×ℝ𝑚}||𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′𝑢^,𝑣^||||𝑢+𝑢^||^𝑏11+2^𝑗|𝑢|^𝑏11+2^{𝑗′∧𝑘′}||𝑣+𝑣^||^𝑏21+2^𝑗∧𝑘|𝑣|^𝑏2≲2_{−𝐿|𝑗−𝑗′|}𝑑𝑢𝑑𝑣2^{−𝐿|𝑘−𝑘′|}𝜃^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦),(3.9)where we have used Lemma 2.2 in the last inequality. Applying Minkowski’s inequality and Hölder’s inequality yields‖‖‖‖‖⎧⎪⎨⎪⎩2_𝑗𝛼2^𝑘𝛽_𝑗′_,𝑘′>0sup_{(𝑢,𝑣)∈ℝ𝑛×ℝ𝑚}||𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′||∗||𝜃_{𝑗′,𝑘′}∗𝑓||(𝑥−𝑢,𝑦−𝑣)1+2^𝑗|𝑢|^𝑏11+2^𝑗∧𝑘|𝑣|^𝑏2⎫⎪⎬⎪⎭_𝑗,𝑘‖‖‖‖‖_ℓ𝑞≲⎧⎪⎨⎪⎩_{𝑗′,𝑘′∈ℤ+}⎛⎜⎜⎝_{𝑗,𝑘∈ℤ+}2^{−|𝑗−𝑗′|𝑞(𝐿−𝜀−|𝛼|)}2^{−|𝑘−𝑘′|𝑞(𝐿−𝜀−|𝛽|)}×2_{𝑗′𝛼𝑞}⎞⎟⎟⎠2^{𝑘′𝛽𝑞}𝜃^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦)^𝑞^1/𝑞≲‖‖‖2^𝑗′𝛼2^𝑘′𝛽𝜃^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)(𝑥,𝑦)_{𝑗′,𝑘′}‖‖‖_ℓ𝑞,(3.10)

where we have chosen 𝜀 as a small positive constant less than 𝐿−(|𝛼|∨|𝛽|). Therefore,𝐼₂≲‖‖‖2^𝑗′𝛼2^𝑘′𝛽𝜃^{∗⃗𝑏,𝑗}_′,𝑘′(𝑓)_{𝑗′,𝑘′}‖‖‖_{𝐿𝑝(ℓ𝑞)}.(3.11) Combining the estimates (3.8) and (3.11), we obtain (3.5). This finishes the proof of (3.1), and hence, Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces follow.

The proofs of Theorems 1.3 and 1.5 for Besov spaces are similar. By (3.2) and the maximal function estimate (2.14), ‖𝜓_𝑗,𝑘∗𝑓‖_𝑝∼‖𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)‖_𝑝. The conclusion of Theorem 1.5 for Besov spaces follows. By Calderón’s reproducing formula (2.1), Young’s inequality, the almost orthogonality estimate in Lemma 2.2, Hölder’s inequality, and Minkowski’s inequality, we have^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝≤2^𝛼𝑞2^𝛽𝑞‖‖𝜓_1,1∗𝜙‖‖^𝑞1‖𝜙∗𝑓‖^𝑞𝑝+^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞⎛⎜⎜⎝^∞_𝑗_′∞=1_𝑘′=1‖‖𝜓_𝑗,𝑘∗𝜃_𝑗′,𝑘^′‖‖₁‖‖𝜃_{𝑗′,𝑘′}∗𝑓‖‖_𝑝⎞⎟⎟⎠^𝑞≲‖𝜙∗𝑓‖^𝑞𝑝+^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞⎛⎜⎜⎝^∞_𝑗_′∞=1_𝑘′=12^{−(𝐿−𝜀)𝑞(|𝑗−𝑗′|+|𝑘−𝑘′|)}‖‖𝜃_{𝑗′𝑘′}∗𝑓‖‖^𝑞𝑝≲‖𝜙∗𝑓‖_𝑞𝑝⎞⎟⎟⎠+^∞_𝑗_′∞=1_𝑘′=12^{𝑗′𝛼𝑞}2^{𝑘′𝛽𝑞}‖‖𝜃_{𝑗′,𝑘′}∗𝑓‖‖^𝑞𝑝,(3.12)

as desired. This ends the proof of Theorem 1.3 for Besov case. Hence, the proofs of Theorems 1.3 and 1.5 are complete.

4. Proof of Theorem 1.8

4.1. Proof of Theorem 1.8 for Besov Space

By the moment conditions of 𝜓^𝑗(1)’s and 𝜓^𝑘(2)’s, we may write𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)=_{ℝ𝑛+𝑚×ℝ𝑚}𝜓^𝑗(1)(𝑢,𝑣)𝜓^𝑘(2)(𝑤)Δ^{𝔉−𝑢,−𝑣;−𝑤}^𝑀𝑓(𝑥,𝑦)𝑑𝑢𝑑𝑣𝑑𝑤.(4.1)

By Minkowski’s inequality and Hölder’s inequality and noting that‖‖(Δ^{𝔉−𝑢,−𝑣;−𝑤})^𝑀𝑓‖‖_𝑝=‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖_𝑝,(4.2)

we have‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝=_{ℝ𝑛+𝑚×ℝ𝑚}||𝜓^𝑗(1)(𝑢,𝑣)||||𝜓^𝑘(2)(𝑤)||‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖_𝑝𝑑𝑢𝑑𝑣𝑑𝑤^𝑞≲_{ℝ𝑛+𝑚×ℝ𝑚}||𝜓^𝑗(1)(𝑢,𝑣)||||𝜓^𝑘(2)(𝑤)||‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖^𝑞𝑝𝑑𝑢𝑑𝑣𝑑𝑤.(4.3)

Hence,^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝≲_{ℝ𝑛+𝑚×ℝ𝑚}^∞_𝑗=12^𝑗𝛼𝑞||𝜓^𝑗(1)(𝑢,𝑣)||^∞_𝑘=1≲‖𝑓‖_{𝑞ℬ𝛼,𝛽𝑝,𝑞,(2)}2^𝑘𝛽𝑞||𝜓^𝑘(2)(𝑤)||‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖^𝑞𝑝𝑑𝑢d𝑣𝑑𝑤,(4.4)

where the last inequality follows from^∞_𝑘=12^𝑘𝛽𝑞||𝜓^𝑘(2)(𝑤)||≲^∞_𝑘=12^𝑘𝛽𝑞2^−𝑘𝐿2^−𝑘+|𝑤|^𝑚+𝐿≲1|𝑤|^{𝑚+𝛽𝑞}_∞,_𝑗=12^𝑗𝛼𝑞||𝜓^𝑗(1)(𝑢,𝑣)||≲1||(𝑢,𝑣)||^{𝑛+𝑚+𝛼𝑞}.(4.5)

This inequality together with the trivial inequality ‖𝑆₀𝑓‖_𝑝≲‖𝑓‖_𝑝 yields ‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞}≲‖𝑓‖_ℬ^{𝛼,𝛽𝑝,𝑞,(2)}.

To prove the converse, by Lemma 2.6, it suffices to show that_{ℝ𝑛+𝑚×ℝ𝑚}⎛⎜⎜⎝‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓‖‖_𝑝||(𝑢,𝑣)||^𝛼|𝑤|^𝛽⎞⎟⎟⎠^𝑞𝑑𝑢𝑑𝑣𝑑𝑤||(𝑢,𝑣)||^𝑛+𝑚|𝑤|^𝑚≲‖𝑓‖^𝑞ℬ_{𝛼,𝛽𝑝,𝑞}.(4.6)By the Calderón’s identity (2.1), we write(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓=Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝜑∗𝑆₀𝑓+^∞_∞𝑗=1_𝑘=1Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘∗𝜓^𝑗,𝑘∗𝑓,(4.7)

where the series on the right hand side converges in 𝐿^𝑝(ℝ^𝑛+𝑚). Thus, by Minkowski’s inequality and Young’s inequality, we conclude that‖‖‖Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝑓‖‖‖_𝑝≤‖‖‖Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝜑‖‖‖₁‖‖𝑆₀𝑓‖‖_𝑝+^∞_∞𝑗=1_𝑘=1‖‖‖Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘‖‖‖₁‖‖𝜓_𝑗,𝑘∗𝑓‖‖_𝑝.(4.8)

It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of𝐼𝐼𝐼₁≡_{𝑗′∈ℤ𝑘}_′∈ℤ2^{𝑗′𝛼𝑞}2^{𝑘′𝛽𝑞}‖‖‖Δ_{𝔉𝑢,𝑣;𝑤}𝐼𝐼𝐼₂^𝑀𝜑‖‖‖^𝑞1‖‖𝑆₀𝑓‖‖^𝑞𝑝,≡∑_{𝑗′∈ℤ}∑_{𝑘′∈ℤ}2^{𝑗′𝛼𝑞}2^{𝑘′𝛽𝑞}^∞_∞∑_𝑗=1∑_𝑘=1‖‖(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝜓_𝑗,𝑘‖‖₁‖‖𝜓_𝑗,𝑘∗𝑓‖‖_𝑝^𝑞.(4.9)

For 𝐼𝐼𝐼₁, we have𝐼𝐼𝐼₁≲⎧⎪⎨⎪⎩_{𝑗′𝑘>0′>0}2^{−𝑗′𝑞(𝑀−𝛼)}2^{−𝑘′𝑞(𝑀−𝛽)}+_{𝑘𝑗>0′≤0}2^{−𝑗′𝑞(𝑀−𝛼)}2^{𝑘′𝑞𝛽}+𝑗_≤0_𝑘′>02^{𝑗′𝑞𝛼}2^{−𝑘′𝑞(𝑀−𝛽)}+_{𝑗′𝑘≤0′≤0}2^{𝑗′𝑞𝛼}2^{𝑘′𝑞𝛼}⎫⎪⎪⎪⎬⎪⎪⎪⎭‖‖𝑆₀𝑓‖‖^𝑞𝑝≲‖‖𝑆₀𝑓‖‖^𝑞𝑝,(4.10)

where we have used the estimate‖‖‖Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜑‖‖‖₁≲2^{𝑀⋅min{0,−𝑗′,−𝑘′,−𝑗′−𝑘′}}.(4.11)

To estimate 𝐼𝐼𝐼₂, by the estimate‖‖‖Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘‖‖‖₁≲2^{𝑀⋅min{0,𝑗−𝑗′,𝑘−𝑘′,𝑗−𝑗′+𝑘−𝑘′}},(4.12)

and Hölder’s inequality, we see that 𝐼𝐼𝐼₂ is majorized by_𝑗′_{,𝑘′∈ℤ}⎛⎜⎜⎜⎜⎝_{0<𝑗≤𝑗′0<𝑘≤𝑘′}2^{𝑞[𝑀−𝜀−(𝛼∨𝛽)](𝑗−𝑗′+𝑘−𝑘′)}+_{0<𝑗≤𝑗′𝑘>(𝑘′∨0)}2^{𝑞[(𝑀−𝛼−𝜀)(𝑗−𝑗′)+(𝑘′−𝑘)(𝛽−𝜀)]}+_{𝑗>(𝑗′∨0)0<𝑘≤𝑘′}2^{𝑞(𝑀−𝛽−𝜀)(𝑘−𝑘′)}2^{𝑞(𝑗′−𝑗)(𝛼−𝜀)}+𝑗>(𝑗_∨0)_{𝑘>(𝑘′∨0)}2^{𝑞(𝑗′−𝑗)(𝛼−𝜀)}2^{𝑞(𝑘′−𝑘)(𝛽−𝜀)}×2_𝑗𝛼𝑞⎞⎟⎟⎟⎟⎟⎟⎠2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝≲_𝑗>0_𝑘>02^𝑗𝛼𝑞2^𝑘𝛽𝑞‖‖𝜓_𝑗,𝑘∗𝑓‖‖^𝑞𝑝,(4.13)

proving (4.6), where 𝜀 is a positive number such that 0<𝜀<(𝛼∧𝛽)<(𝛼∨𝛽)+𝜀<𝑀. This concludes the proof of Theorem 1.8 for Besov spaces.

4.2. Proof of Theorem 1.8 for Triebel-Lizorkin Space

Using the moment conditions on 𝜓_𝑗,𝑘, we have_𝑗,𝑘_∈ℤ+2^𝑗𝛼𝑞2^𝑘𝛽𝑞||𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)||^𝑞≤_{𝑗,𝑘∈ℤ+}2^𝑗𝛼𝑞2^𝑘𝛽𝑞_{ℝ𝑛+𝑚×ℝ𝑚}≤||𝜓^𝑗(1)(𝑢,𝑣)||||𝜓^𝑘(2)(𝑤)|||||(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓(𝑥,𝑦)|||^𝑞𝑑𝑢𝑑𝑣𝑑𝑤_{ℝ𝑛+𝑚×ℝ𝑚}^∞_𝑗=12^𝑗𝛼𝑞||𝜓^𝑗(1)(𝑢,𝑣)||^∞_𝑘=1≲2^𝑘𝛽𝑞||𝜓^𝑘(2)(𝑤)|||||Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝑓(𝑥,𝑦)|||^𝑞𝑑𝑢𝑑𝑣𝑑𝑤_{ℝ𝑛+𝑚×ℝ𝑚}⎛⎜⎜⎜⎝|||Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝑓(𝑥,𝑦)||||(𝑢,𝑣)|^𝛼|𝑤|^𝛽⎞⎟⎟⎟⎠^𝑞𝑑𝑢𝑑𝑣𝑑𝑤||(𝑢,𝑣)||^𝑛+𝑚|𝑤|^𝑚,(4.14)

where 𝑔̃(𝑥)∶=𝑔(−𝑥). Therefore,‖‖‖‖^∞_∞𝑗=1_𝑘=12^𝑗𝛼𝑞2^𝑘𝛽𝑞||𝜓_𝑗,𝑘∗𝑓||^𝑞^1/𝑞‖‖‖‖_𝑝≲‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞,(2)},(4.15)

which, together with the obvious inequality ‖𝑆₀𝑓‖_𝑝≲‖𝑓‖_𝑝, yields ‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}≲‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞,(2)}.

To show the converse, writeΔ^{𝔉𝑢,𝑣;𝑤}^𝑀𝑓=Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝜑∗S₀𝑓+^∞_∞𝑗=1_𝑘=1Δ^{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘∗𝜓^𝑗,𝑘∗𝑓,(4.16)

where the series converges in 𝒮^𝔉(ℝ^𝑛×ℝ^𝑚). It follows that‖‖‖‖_{ℝ𝑛+𝑚×ℝ𝑚}|(Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝑓||(𝑢,𝑣)|^𝛼|𝑤|^𝛽^𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|^𝑛+𝑚|𝑤|^𝑚^1/𝑞‖‖‖‖_𝑝≤‖‖‖‖_{𝑗∈ℤ𝑘}_∈ℤ2^{𝑗𝛼𝑞}2^{𝑘𝛽𝑞}|((Δ^{𝔉𝑢,𝑣;𝑤})^𝑀𝜑)∗𝑆₀𝑓|^𝑞^1/𝑞‖‖‖‖_𝑝+‖‖‖‖_{𝑗∈ℤ𝑘}_∈ℤ2^{𝑗𝛼𝑞}2^{𝑘𝛽𝑞}^∞_∞𝑗=1_𝑘=1||||Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘∗𝜓^𝑗,𝑘∗𝑓||||^𝑞^1/𝑞‖‖‖‖_𝑝∶=𝐼𝑉₁+𝐼𝑉₂,(4.17)where |(𝑢,𝑣)|∼2^−𝑗′ and |𝑤|∼2^−𝑘′.

For 𝐼𝑉₂, by Lemma 2.2,|||Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘(𝑢,𝑣)|||≲2^{𝑀⋅min{0,𝑗−𝑗′,𝑘−𝑘′,𝑗−𝑗′+𝑘−𝑘′}}2^𝑗𝑛2^{(𝑗∧𝑘)𝑚}1+2^𝑗|𝑢|+2^𝑗∧𝑘|𝑣|^𝐿2.(4.18)

Consequently,|||Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘∗𝜓_𝑗,𝑘∗𝑓(𝑥,𝑦)|||≲2^{𝑀⋅min{0,𝑗−𝑗′,𝑘−𝑘′,𝑗−𝑗′+𝑘−𝑘′}}𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦).(4.19)Therefore,_𝑗_′∈ℤ_{𝑘′∈ℤ}2^{𝑗′𝛼𝑞}2^{𝑘′𝛽𝑞}^∞_∞𝑗=1_𝑘=1||||Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜓_𝑗,𝑘∗𝜓^𝑗,𝑘∗𝑓(𝑥,𝑦)||||^𝑞≲_{𝑗,𝑘∈ℤ+}⎛⎜⎜⎜⎝_{𝑗′𝑘>𝑗′>𝑘}2^{𝑞[𝑀−𝜀−(𝛼∨𝛽)](𝑗−𝑗′+𝑘−𝑘′)}+_{𝑗′𝑘>𝑗′<𝑘}2^{𝑞(𝑀−𝛼−𝜀)(𝑗−𝑗′)}2^{𝑞(𝑘′−𝑘)(𝛽−𝜀)}+_{𝑗′𝑘<𝑗′>𝑘}2^{𝑞(𝑀−𝛽−𝜀)(𝑘−𝑘′)}2^{𝑞(𝑗′−𝑗)(𝛼−𝜀)}+_{𝑗′𝑘<𝑗′<𝑘}2^{𝑞(𝑗′−𝑗)(𝛼−𝜀)}2^{𝑞(𝑘′−𝑘)(𝛽−𝜀)}×2_𝑗𝛼𝑞⎞⎟⎟⎟⎠2^𝑘𝛽𝑞𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦)^𝑞≲_{𝑗,𝑘∈ℤ+}2^𝑗𝛼𝑞2^𝑘𝛽𝑞𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)(𝑥,𝑦)^𝑞,(4.20)

which together with the maximal function characterization of ℱ^{𝛼,𝛽𝑝,𝑞} implies𝐼𝑉₂≲‖‖‖‖‖⎧⎪⎨⎪⎩_{𝑗,𝑘∈ℤ+}2^𝑗𝛼𝑞2^𝑘𝛽𝑞𝜓^{∗⃗𝑏,𝑗,𝑘}(𝑓)^𝑞⎫⎪⎬⎪⎭^1/𝑞‖‖‖‖‖_𝑝≲‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}.(4.21)

As for 𝐼𝑉₁, similar estimate to (4.19) yields|||Δ_{𝔉𝑢,𝑣;𝑤}^𝑀𝜑∗𝑆₀𝑓(𝑥,𝑦)|||≲2^{−𝑀⋅max{0,𝑗′,𝑘′,𝑗′+𝑘′}}𝜓^{∗⃗𝑏,0,0}(𝑓)(𝑥,𝑦).(4.22)

Therefore,𝐼𝑉₁≲⎧⎪⎨⎪⎩_{𝑗′>0𝑘>0}2^{−𝑗′𝑞(𝑀−𝛼)}2^{−𝑘′𝑞(𝑀−𝛽)}+_{𝑗>0𝑘≤0}2^{−𝑗′𝑞(𝑀−𝛼)}2^{𝑘′𝑞𝛽}+_{𝑗′𝑘≤0′>0}2^{𝑗′𝑞𝛼}2^{−𝑘′𝑞(𝑀−𝛽)}+_{𝑗′𝑘≤0′≤0}2^{𝑗′𝑞𝛼}2^{𝑘′𝑞𝛼}⎫⎪⎪⎬⎪⎪⎭^1/𝑞‖‖𝜓^{∗⃗𝑏,0,0}(𝑓)‖‖_𝑝≲‖‖𝑆₀𝑓‖‖_𝑝.(4.23)

This estimate together with (4.21) and Lemma 2.6 yields ‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞,(2)}≲‖𝑓‖_ℱ^{𝛼,𝛽𝑝,𝑞}. This ends the proof of Theorem 1.8.

Acknowledgments

The research was supported by NNSF of China (Grant nos. 11101423, 11171345) and the Fundamental Research Funds for the Central Universities of China (Grant no. 2009QS12). The authors would like to express their deep gratitude to the referee for his/her valuable comments and suggestions.