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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 275791, 18 pages
http://dx.doi.org/10.1155/2012/275791
Research Article

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

Department of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China

Received 25 February 2012; Accepted 9 April 2012

Academic Editor: Yongsheng S. Han

Copyright © 2012 Xinfeng Wu and Zongguang Liu. 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.

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 𝑛, in 2001, Nagel et al. [2] studied a class of product singular integrals with flag kernel. They proved, among other things, the 𝐿𝑝 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<𝑝1, Han and Lu [7] developed Hardy spaces 𝐻𝑝𝔉(𝑛×𝑚) with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calderón’s identity and proved the 𝐻𝑝𝔉(𝑛×𝑚)𝐻𝑝𝔉(𝑛×𝑚) and 𝐻𝑝𝔉(𝑛×𝑚)𝐿𝑝(𝑛×𝑚) boundedness for flag singular integral operators. The duality of 𝐻𝑝𝔉(𝑛×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(𝑛×𝑚) with𝜓supp(1)(𝜉,𝜂)𝑛×𝑚12||||,𝜉,𝜂2𝑗|||𝜓(1)2𝑗𝜉,2𝑗𝜂|||2=1(𝜉,𝜂)𝑛×𝑚{(0,0)},(1.1)

and let 𝜓(2)𝒮(𝑚) with𝜓supp(2)𝜂𝑚12||𝜂||,2𝑘|||𝜓(2)2𝑘𝜂|||2=1𝜂𝑚{0}.(1.2)

Let 𝜓𝑗(1)(𝑥,𝑦)=2𝑗𝑛𝜓(1)(2𝑗𝑥,2𝑗𝑦),𝜓𝑘(2)(𝑦)=2𝑘𝑚𝜓(2)(2𝑘𝑦), and𝜓𝑗,𝑘(𝑥,𝑦)=𝜓𝑗(1)2𝜓𝑘(2)(𝜓𝑥,𝑦)=𝑗(1)(𝑥,𝑦𝑧)𝜓𝑘(2)(𝑧)𝑑z,(1.3)

then, for 𝑓𝐿2(𝑛×𝑚), the following Calderón’s reproducing formula holds:𝑓=𝑗𝑘𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓,(1.4)

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

A Schwartz function 𝑓𝒮(𝑛+2𝑚) is said to be a product test function in 𝒮(𝑛+𝑚×𝑚) if it satisfies𝑚×𝑛𝑓(𝑥,𝑦,𝑧)𝑥𝛼𝑦𝛽=𝑑𝑥𝑑𝑦𝑚𝑓(𝑥,𝑦,𝑧)𝑧𝛾𝑑𝑧=0,(1.5)

for all multi-indices 𝛼𝑛,𝛽,𝛾𝑚.

Definition 1.1. A function 𝑓(𝑥,𝑦) defined on 𝑛×𝑚 is said to be a test function in 𝒮𝔉(𝑛×𝑚) if there exists a function 𝑓#𝒮(𝑚+𝑛×𝑚) such that 𝑓(𝑥,𝑦)=𝑚𝑓#(𝑥,𝑦𝑧,𝑧)𝑑𝑧,(1.6)and the seminorm of 𝑓 is defined by 𝑓𝔉,𝛼,𝛽𝑓=inf#𝛼,𝛽forallrepresentationsof𝑓in(1.7),(1.7) where 𝛼,𝛽 denote the seminorm in 𝒮(𝑛+2𝑚). Denote by 𝒮𝔉(𝑛×𝑚) the dual of 𝒮𝔉(𝑛×𝑚).

Choose a Schwartz function 𝜑 on 𝑛×𝑚 such that||||𝜑(𝜉,𝜂)2=1𝑗=1𝑘=1|||𝜓(1)2𝑗𝜉1,2𝑗𝜉2|||2|||𝜓(2)2𝑘𝜉2|||2.(1.8)Note that 𝜑𝒮𝔉(𝑛×𝑚) with Fourier transform is supported in {(𝜉,𝜂)|(𝜉,𝜂)|1}. Define the operator 𝑆0 by 𝑆0𝑓=𝜑𝑓,𝑓𝒮𝔉(𝑛×𝑚).

For 𝑓𝐿2(𝑛×𝑚), by taking the Fourier transform,𝑓=𝜑𝜑𝑓+𝑗=1𝑘=1𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓,(1.9)

where the series converges in 𝐿2(𝑛×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 𝛼,𝛽𝑝,𝑞(𝑛×𝑚) is defined to be the collection of all 𝑓𝒮𝔉(𝑛×𝑚) such that 𝑓𝜓𝛼,𝛽𝑝,𝑞(𝑛×𝑚)𝑆=0𝑓𝐿𝑝(𝑛×𝑚)+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞||𝜓𝑗,𝑘||𝑓𝑞1/𝑞𝐿𝑝(𝑛×𝑚)<,(1.10) and the inhomogeneous Besov space associated with flag singular integrals 𝛼,𝛽𝑝,𝑞(𝑛×𝑚) is defined to be the collection of all 𝑓𝒮𝔉(𝑛×𝑚) such that 𝑓𝜓𝛼,𝛽𝑝,𝑞(𝑛×𝑚)𝑆=0𝑓𝐿𝑝(𝑛×𝑚)+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝐿𝑝(𝑛×𝑚)1/𝑞<.(1.11)

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

Theorem 1.3. If 𝜃𝑗,𝑘 satisfies the same conditions as 𝜓𝑗,𝑘, and 𝜙 is defined similar to (1.8) with 𝜓 replaced by 𝜃, then for 𝛼,𝛽 and 𝑝,𝑞(1,) and 𝑓𝒮𝔉, 𝑓𝜃𝛼,𝛽𝑝,𝑞𝑓𝜓𝛼,𝛽𝑝,𝑞,𝑓𝜃𝛼,𝛽𝑝,𝑞𝑓𝜓𝛼,𝛽𝑝,𝑞.(1.12)

Remark 1.4. As the classical case, it is not hard to show that 𝜓𝛼,𝛽𝑝,𝑞 and 𝜓𝛼,𝛽𝑝,𝑞 are norms of 𝛼,𝛽𝑝,𝑞 and 𝛼,𝛽𝑝,𝑞, respectively. Moreover, 𝛼,𝛽𝑝,𝑞 and 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 𝑗𝑘=min{𝑗,𝑘} and 𝑗𝑘=max{𝑗,𝑘}. We introduce the following flag multi-parameter Peetre maximal functions (with respect to 𝜓). For 𝑗,𝑘+ and 𝑏=(𝑏1,𝑏2)+×+, define𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)=sup(𝑢,𝑣)𝑛×𝑚||𝜓𝑗,𝑘||𝑓(𝑥𝑢,𝑦𝑣)(1+2𝑗|𝑢|)𝑏1(1+2𝑗𝑘|𝑣|)𝑏2.(1.13)

For 𝑗=𝑘=0, define𝜑𝑏,0,0(𝑓)(𝑥)=sup𝑦𝑛||||𝜑𝑓(𝑥𝑢,𝑦𝑣)(1+|𝑢|)𝑏1(1+|𝑣|)𝑏2.(1.14)

An index 𝑏=(𝑏1,𝑏2) is said to be admissible if𝑏1>𝑛𝑝𝑞,𝑏2>𝑚.𝑝𝑞(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 and 𝑝,𝑞(1,). If 𝑏 is admissible, then for 𝑓𝒮𝔉, one has(i)𝑓𝛼,𝛽𝑝,𝑞,(1)=𝑆0𝑓𝑝+{2𝛼𝑗2𝛽𝑘𝜓𝑏,𝑗,𝑘(𝑓)𝑝}𝑗,𝑘𝑞𝑓𝛼,𝛽𝑝,𝑞, (ii)𝑓𝛼,𝛽𝑝,𝑞,(1)=𝑆0𝑓𝑝+{2𝛼𝑗2𝛽𝑘𝜓𝑏,𝑗,𝑘(𝑓)}𝑗,𝑘𝑞𝐿𝑝𝑓𝛼,𝛽𝑝,𝑞. Here and in what follows, one uses the following notation: {𝑎𝑗,𝑘}𝑗,𝑘𝑞=𝑗=1𝑘=1|𝑎𝑗,𝑘|𝑞1/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 𝑑. A 𝑘-normalized bump function on a space 𝑑 is a 𝐶𝑘 function supported on the unit ball with 𝐶𝑘 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 𝑛×𝑚 is a distribution 𝐾 on 𝑛+𝑚 which coincides with a 𝐶 function away from the coordinate subspaces (0,𝑦), where (0,𝑦)𝑛×𝑚 and satisfies(1) (Differential inequalities) for any multi-indices 𝛼 and  𝛽, ||𝜕𝛼𝑥𝜕𝛽𝑦||𝐾(𝑥,𝑦)𝐶𝛼,𝛽|𝑥|𝑛|𝛼|(|𝑥|+|𝑦|)𝑚|𝛽|,(1.17) for all (𝑥,𝑦)𝑛×𝑚 with |𝑥|0,(2) (Cancellation condition) ||||𝑚𝜕𝛼𝑥𝐾(𝑥,𝑦)𝜙1||||(𝛿𝑦)𝑑𝑦𝐶𝛼|𝑥|𝑛|𝛼|,(1.18) for all multi-index 𝛼 and every normalized bump function 𝜙1 on 𝑚 and every 𝛿>0, ||||𝑛𝜕𝛽𝑦𝐾(𝑥,𝑦)𝜙2||||(𝛿𝑥)𝑑𝑥𝐶𝛽|𝑦|𝑚|𝛽|(1.19) for every multi-index 𝛽 and every normalized bump function 𝜙2 on 𝑛 and every 𝛿>0, ||||𝑛+𝑚𝐾(𝑥,𝑦)𝜙3𝛿1𝑥,𝛿2𝑦||||𝑑𝑥𝑑𝑦𝐶,(1.20) for every normalized bump function 𝜙3 on +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 𝑛×𝑚 with the flag kernel 𝐾, then, for 𝑝,𝑞>1 and 𝛼,𝛽, 𝑇 is bounded on 𝛼,𝛽𝑝,𝑞 and on 𝛼,𝛽𝑝,𝑞.

As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will give the difference characterization for 𝛼,𝛽𝑝,𝑞 and 𝛼,𝛽𝑝,𝑞. However, the new feature is that the differences of functions are associated with the “flag.’’ More precisely, for (𝑢,𝑣)𝑛×𝑚 and 𝑤𝑚, we define the first flag difference (associated the flag {(0,0)}{(0,𝑦)}) in 𝑛×𝑚 byΔ𝔉𝑢,𝑣;𝑤Δ𝑓(𝑥,𝑦)=𝑢,𝑣Δ𝑤(2)𝑓(𝑥,𝑦)=𝑓(𝑥+𝑢,𝑦+𝑣+𝑤)𝑓(𝑥+𝑢,𝑦+𝑣)𝑓(𝑥,𝑦+𝑤)+𝑓(𝑥,𝑦),(1.21)

where Δ𝑢,𝑣 is the difference operator on 𝑚+𝑛, and Δ𝑤(2) is the difference operator on 𝑚. For 𝑘+ and 𝑘2, the 𝑘th flag difference operator (Δ𝔉𝑢,𝑣;𝑤)𝑘 can be defined inductively by(Δ𝔉𝑢,𝑣;𝑤)𝑘=Δ𝔉𝑢,𝑣;𝑤(Δ𝔉𝑢,𝑣;𝑤)𝑘1.(1.22)

Theorem 1.8. If 𝛼,𝛽>0, 1<𝑝<, 1<𝑞<, and 𝑀[𝛼𝛽]+1, where [] denotes the greatest integer function, one defines 𝑓𝛼,𝛽𝑝,𝑞,(2)=𝑓𝑝+𝑛+𝑚×𝑚(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑝|(𝑢,𝑣)|𝛼|𝑤|𝛽𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|𝑛+𝑚|𝑤|𝑚1/𝑞,𝑓𝛼,𝛽𝑝,𝑞,(2)=𝑓𝑝+𝑛+𝑚×𝑚|(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓||(𝑢,𝑣)|𝛼|𝑤|𝛽𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|𝑛+𝑚|𝑤|𝑚1/𝑞𝑝,(1.23) then 𝑓𝛼,𝛽𝑝,𝑞,(2)𝑓𝛼,𝛽𝑝,𝑞,𝑓𝛼,𝛽𝑝,𝑞,(2)𝑓𝛼,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 𝒮𝔉

Lemma 2.1. The inhomogeneous Calderón’s reproducing formula holds 𝑓=𝜑𝜑𝑓+𝑗=1𝑘=1𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓,(2.1) where the series converges in 𝐿𝑝(𝑛+𝑚)(1<𝑝<), 𝒮𝔉(𝑛×𝑚) and 𝒮𝔉(𝑛×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 𝑓𝒮𝔉(𝑛×𝑚), then by definition, there exists 𝑓#𝒮(𝑛+𝑚×𝑚) such that 𝑓(𝑥,𝑦)=𝑚𝑓#(𝑥,𝑦𝑧,𝑧)𝑑𝑧. We need to show that for all 𝑁+, 𝜑#𝜑#𝑓#+𝑗,𝑘+𝑗𝑁or𝑘𝑁𝜓#𝑗,𝑘𝜓#𝑗,𝑘𝑓#,(2.2) tends to 𝑓 in the topology of 𝒮(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#𝑁𝑓(𝑥,𝑦)=#𝑓#𝑁(𝑥,𝑦),forevery(𝑥,𝑦)𝑛+𝑚×𝑚.(2.3) Let 𝑁(𝜉,𝜂,𝑥,𝑦)=𝑒𝑖(𝑥𝜉+𝑦𝜂)𝑓#(𝜉,𝜂)𝑘=𝑁+1|𝜓(2)(2𝑘𝜂)|2 so that 𝑓#(𝑥,𝑦)𝑓#𝑁(𝑥,𝑦)=𝑐𝑛𝑛+𝑚×𝑚𝑁(𝜉,𝜂,𝑥,𝑦)𝑑𝜂𝑑𝜉.(2.4) Since |𝑁𝑓(𝜉,𝜂,𝑥,𝑦)||#(𝜉,𝜂)|𝐿1(𝑛+𝑚×𝑚), Lebesgue’s dominated convergence theorem yields lim𝑁+𝑓#𝑁(𝑥,𝑦)=𝑓#(𝑥,𝑦),forevery(𝑥,𝑦)𝑛+𝑚×𝑚.(2.5)
On the other hand, using the cancellation conditions of 𝜓𝑗(1)𝜓𝑗(1),𝜓𝑘(2)𝜓𝑘(2), and the smoothness of 𝑓#, we can get ||𝜓#𝑗,𝑘𝜓#𝑗,𝑘𝑓#||(𝑥,𝑦)2𝑗2𝑘||𝑦||1+|𝑥|+𝐿,𝑗,𝑘,𝐿+.(2.6)
Now, (2.5) together with the estimate (2.6) implies that, for 𝑓𝒮𝔉(𝑛×𝑚), ||𝑓#(𝑥)𝑓#𝑁||=(𝑥,𝑦)𝑗+𝑘𝑁+1||𝜓#𝑗,𝑘𝜓#𝑗,𝑘𝑓#||(𝑥,𝑦)2𝑁||𝑦||1+|𝑥|+𝐿.(2.7) Applying this to 𝜕𝑢𝑓# (here 𝑢 denotes any multi-index in 𝑛+2𝑚) and noting that 𝜕𝑢𝑓#𝑁=(𝜕𝑢𝑓#)𝑁, we obtain lim𝑁+sup(𝑥,𝑦)𝑛×𝑚||𝑦||)(1+|𝑥|+𝐿||𝜕𝑢𝑓#𝑓#𝑁||(𝑥,𝑦)=0.(2.8) This proves the convergence of series in (2.1) in 𝒮𝔉(𝑛×𝑚). The convergence in 𝒮𝔉(𝑛×𝑚) follows by a standard duality argument. The convergence in 𝐿𝑝(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 𝑥𝑛,𝑦𝑚. Given any positive integers 𝐿 and 𝑀, there exists a constant 𝐶=𝐶(𝐿,𝑀)>0 such that ||𝜓𝑗,𝑘𝜑𝑗,𝑘||(𝑥,𝑦)𝐶2|𝑗𝑗|𝐿2|𝑘𝑘|𝐿2(𝑗𝑗)𝑀(2𝑗𝑗+|𝑥|)𝑛+𝑀2(𝑗𝑗𝑘𝑘)𝑀(2𝑗𝑗𝑘𝑘+||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+, and for any 𝐿>0 and 𝑏=(𝑏1,𝑏2)+×+, there exists a constant 𝐶=𝐶(𝐿,𝑏) depending only on 𝐿 and 𝑏, but independent on 𝑗,𝑘,𝑗,𝑘, such that ||𝜓𝑗,𝑘𝜓𝑗,𝑘𝜓𝑗,𝑘(||𝑓𝑥,𝑦)𝐶2𝐿|𝑗𝑗|2𝐿|𝑘𝑘|𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦).(2.10)

Proof. By the almost orthogonality estimate in Lemma 2.2, for any 𝐿>0 and 𝑀>𝑏1𝑏2, we have the pointwise estimate ||𝜓𝑗,𝑘𝜓𝑗,𝑘𝜓𝑗,𝑘||𝑓(𝑥,𝑦)2|𝑗𝑗|𝐿2|𝑘𝑘|𝐿×𝑛×𝑚2(𝑗𝑗)𝑛2(𝑗𝑘𝑗𝑘)𝑚||𝜓𝑗,𝑘||𝑓(𝑥𝑢,𝑦𝑣)1+2𝑗𝑗|𝑢|𝑀+𝑛1+2𝑗𝑘𝑗𝑘|𝑣|𝑀+𝑚𝑑𝑢𝑑𝑣2|𝑗𝑗|𝐿2|𝑘𝑘|𝐿𝜓𝑏,𝑗,𝑘×(𝑓)(𝑥)𝑛×𝑚2(𝑗𝑗)𝑛2(𝑗𝑘𝑗𝑘)𝑚(1+2𝑗|𝑢|)𝑏1(1+2𝑗𝑘|𝑣|)𝑏21+2𝑗𝑗|𝑢|𝑀+𝑛(1+2𝑗𝑘𝑗𝑘|𝑣|)𝑀+𝑚𝑑𝑢𝑑𝑣2|𝑗𝑗|(𝐿𝑏1)2|𝑘𝑘|(𝐿𝑏1𝑏2)𝜓𝑏,𝑗,𝑘×(𝑓)(𝑥,𝑦)𝑛×𝑚2(𝑗𝑗)𝑛2(𝑗𝑘𝑗𝑘)𝑚1+2𝑗𝑗|𝑢|𝑀+𝑛𝑏1(1+2𝑗𝑘𝑗𝑘|𝑣|)𝑀+𝑚𝑏2𝑑𝑢𝑑𝑣2|𝑗𝑗|(𝐿)2|𝑘𝑘|(𝐿)𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦),(2.11) where 𝐿=𝐿𝑏1𝑏2. This proves (2.10).

Remark 2.4. Since the almost orthogonality estimates hold with 𝜓0,0 or 𝜃0,0 is replaced by 𝜑, repeating the same argument as (2.11), we see that the estimate (2.10) is still valid if 𝜓0,0 or 𝜃0,0 is replaced by 𝜑.

Denote by 𝑠 the strong maximal operator defined by𝑠(𝑓)(𝑥)=sup𝑅𝑥𝑅||||𝑓(𝑦)𝑑𝑦,(2.12)

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

Lemma 2.5. Let 0<𝑐1,𝑐2<, and 0<𝑟<, then for all 𝑗,𝑘 and for all 𝐶1 functions 𝑢 on 𝑛×𝑚 whose Fourier transform is supported in the rectangle {𝜉|𝜉|𝑐12𝑗,|𝜉𝑛|𝑐22𝑗𝑘}, one has the estimate sup𝑦𝑛||||𝑢(𝑥𝑢,𝑦𝑣)(1+2𝑗|𝑢|)𝑛/𝑟(1+2𝑗𝑘|𝑣|)𝑚/𝑟𝐶𝑠(|𝑢|𝑟)(𝑥,𝑦)1/𝑟.(2.13) In particular, if 𝑏=(𝑏1,𝑏2) with 𝑏1𝑛/𝑟 and 𝑏2𝑚/𝑟, then for all 𝑓𝐿𝑝(𝑛×𝑚)(1<𝑝<), 𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑠||𝜓𝑗,𝑘||𝑓𝑟(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: 𝛼,𝛽𝑝,𝑞𝐿𝑝,𝛼,𝛽𝑝,𝑞𝐿𝑝.(2.15)

Proof. For 𝑓𝒮𝔉(𝑛×𝑚), by inhomogeneous Calderón’s reproducing formula (2.1), 𝑓𝑝𝑆0𝑓𝑝+𝑗=1𝑘=1𝜓𝑗,𝑘𝑓𝑝𝑆0𝑓𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝑝1/𝑞𝑓𝛼,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 |𝜓𝑗,𝑘𝑓(𝑥,𝑦)|𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦), Hölder’s inequality, and Fefferman-Stein’s vector-valued inequality, we have 𝑓𝑝𝜑(𝑆0𝑓)𝑝+𝑗=1𝑘=1|𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓|𝑝𝑆0𝑓𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑏,𝑗,𝑘(𝑓)𝑞1/𝑞𝑝𝑆0𝑓𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞|𝜓𝑗,𝑘𝑓|𝑞1/𝑞𝑝𝑓𝛼,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 showing𝑓𝜓𝛼,𝛽𝑝,𝑞𝑓𝜓𝛼,𝛽𝑝,,𝑞,(1)𝑓𝜃𝛼,𝛽𝑝,,𝑞,(1)𝑓𝜃𝛼,𝛽𝑝,𝑞.(3.1)

The first inequality in (3.1) follows from the pointwise inequality||𝜓𝑗,𝑘||𝑓(𝑥,𝑦)𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦),𝑗,𝑘+.(3.2)

Next, for any admissible 𝑏, fix 𝑏. Since 𝑏 is admissible, we can choose 𝑟𝑝𝑞 such that 𝑏1𝑛/𝑟, 𝑏2𝑚/𝑟, and the thus inequality (2.14) holds. We apply Lemma 2.5 and the 𝐿𝑝/𝑟(𝑞/𝑟) boundedness of 𝑠 to deduce𝑓𝜃𝛼,𝛽𝑝,𝑞,(1)2𝑗𝛼2𝑘𝛽𝜃𝑏,𝑗,𝑘(𝑓)𝑗,𝑘𝐿𝑝(𝑞)2𝑗𝛼2𝑘𝛽𝑠(|𝜃𝑗,𝑘𝑓|𝑟)1/𝑟𝑗,𝑘𝐿𝑝(𝑞)𝑓𝜃𝛼,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+, 𝜙𝜃𝑗,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 that𝜑𝑓𝑝𝜑𝜙1𝜙𝑓𝑝+𝜃𝑏,1,1(𝑓)𝑝𝜙𝑓𝑝+2𝑗𝛼2𝑘𝛽𝜃𝑏,𝑗,𝑘(𝑓)𝑗,𝑘𝐿𝑝(𝑞)𝑓𝜃𝛼,𝛽𝑝,𝑞,(1).(3.4)

To finish the proof, it remains to show2𝑗𝛼2𝑘𝛽𝜓𝑏,𝑗,𝑘(𝑓)𝑗,𝑘𝐿𝑝(𝑞)𝑓𝜃𝛼,𝛽𝑝,𝑞,(1).(3.5)

By the inhomogeneous Calderón’s identity (2.1), we have||𝜓𝑗,𝑘||||𝜓𝑓(𝑥,𝑦)𝑗,𝑘||||||(𝜙𝜙𝑓𝑥,𝑦)+𝑗,𝑘+||𝜓𝑗,𝑘𝜃𝑗,𝑘||||𝜃𝑗,𝑘||(𝑓𝑥,𝑦).(3.6)

It follows that2𝑗𝛼2𝑘𝛽𝜓𝑏,𝑗,𝑘(𝑓)𝑗,𝑘𝐿𝑝(𝑞)2𝑗𝛼2𝑘𝛽sup(𝑢,𝑣)𝑛×𝑚(|𝜓𝑗,𝑘𝜙||𝜙𝑓|)(𝑢,𝑣)(1+2𝑗|𝑢|)𝑏1(1+2𝑗𝑘|𝑣|)𝑏2𝑗,𝑘𝐿𝑝(𝑞)+2𝑗𝛼2𝑘𝛽𝑗,𝑘>0sup(𝑢,𝑣)𝑛×𝑚(|𝜓𝑗,𝑘𝜃𝑗,𝑘||𝜃𝑗,𝑘𝑓|)(𝑢,𝑣)(1+2𝑗|𝑢|)𝑏1(1+2𝑗𝑘|𝑣|)𝑏2𝑗,𝑘𝐿𝑝(𝑞)=𝐼1+𝐼2.(3.7)

We first estimate 𝐼1. By the support properties of 𝜓𝑗,𝑘 and 𝜑 and Young’s inequality,𝐼1𝜓1,1𝜙1𝜙𝑓𝑝𝜙𝑓𝑝.(3.8)

Next, we give the estimate for 𝐼2. For any (𝑢,𝑣)𝑛×𝑚,||𝜓𝑗,𝑘𝜃𝑗,𝑘||||𝜃𝑗,𝑘||𝑓(𝑥𝑢,𝑦𝑣)1+2𝑗|𝑢|𝑏11+2𝑗𝑘|𝑣|𝑏2𝑛×𝑚||𝜓𝑗,𝑘𝜃𝑗,𝑘𝑢,𝑣||||𝜃𝑗,𝑘𝑓𝑥𝑢𝑢,𝑦𝑣𝑣||1+2𝑗|𝑢|𝑏11+2𝑗𝑘|𝑣|𝑏2𝑑𝑢𝑑𝑣𝜃𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑛×𝑚||𝜓𝑗,𝑘𝜃𝑗,𝑘𝑢,𝑣||×1+2𝑗||𝑢+𝑢||𝑏11+2𝑗|𝑢|𝑏11+2𝑗𝑘||𝑣+𝑣||𝑏21+2𝑗𝑘|𝑣|𝑏2𝑑𝑢𝑑𝑣2𝐿|𝑗𝑗|2𝐿|𝑘𝑘|𝜃𝑏,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 yields2𝑗𝛼2𝑘𝛽𝑗,𝑘>0sup(𝑢,𝑣)𝑛×𝑚||𝜓𝑗,𝑘𝜃𝑗,𝑘||||𝜃𝑗,𝑘||𝑓(𝑥𝑢,𝑦𝑣)1+2𝑗|𝑢|𝑏11+2𝑗𝑘|𝑣|𝑏2𝑗,𝑘𝑞𝑗,𝑘+𝑗,𝑘+2|𝑗𝑗|𝑞(𝐿𝜀|𝛼|)2|𝑘𝑘|𝑞(𝐿𝜀|𝛽|)×2𝑗𝛼𝑞2𝑘𝛽𝑞𝜃𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑞1/𝑞2𝑗𝛼2𝑘𝛽𝜃𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑗,𝑘𝑞,(3.10)

where we have chosen 𝜀 as a small positive constant less than 𝐿(|𝛼||𝛽|). Therefore,𝐼22𝑗𝛼2𝑘𝛽𝜃𝑏,𝑗,𝑘(𝑓)𝑗,𝑘𝐿𝑝(𝑞).(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), 𝜓𝑗,𝑘𝑓𝑝𝜓𝑏,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 have𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝑝2𝛼𝑞2𝛽𝑞𝜓1,1𝜙𝑞1𝜙𝑓𝑞𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝑗=1𝑘=1𝜓𝑗,𝑘𝜃𝑗,𝑘1𝜃𝑗,𝑘𝑓𝑝𝑞𝜙𝑓𝑞𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝑗=1𝑘=12(𝐿𝜀)𝑞(|𝑗𝑗|+|𝑘𝑘|)𝜃𝑗𝑘𝑓𝑞𝑝𝜙𝑓𝑞𝑝+𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜃𝑗,𝑘𝑓𝑞𝑝,(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 𝜓𝑗(1)’s and 𝜓𝑘(2)’s, we may write𝜓𝑗,𝑘𝑓(𝑥,𝑦)=𝑛+𝑚×𝑚𝜓𝑗(1)(𝑢,𝑣)𝜓𝑘(2)Δ(𝑤)𝔉𝑢,𝑣;𝑤𝑀𝑓(𝑥,𝑦)𝑑𝑢𝑑𝑣𝑑𝑤.(4.1)

By Minkowski’s inequality and Hölder’s inequality and noting that(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑝=(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑝,(4.2)

we have𝜓𝑗,𝑘𝑓𝑞𝑝=𝑛+𝑚×𝑚||𝜓𝑗(1)||||𝜓(𝑢,𝑣)𝑘(2)||(𝑤)(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑝𝑑𝑢𝑑𝑣𝑑𝑤𝑞𝑛+𝑚×𝑚||𝜓𝑗(1)||||𝜓(𝑢,𝑣)𝑘(2)||(𝑤)(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑞𝑝𝑑𝑢𝑑𝑣𝑑𝑤.(4.3)

Hence,𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝑝𝑛+𝑚×𝑚𝑗=12𝑗𝛼𝑞||𝜓𝑗(1)||(𝑢,𝑣)𝑘=12𝑘𝛽𝑞||𝜓𝑘(2)||(𝑤)(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑞𝑝𝑑𝑢d𝑣𝑑𝑤𝑓𝑞𝛼,𝛽𝑝,𝑞,(2),(4.4)

where the last inequality follows from𝑘=12𝑘𝛽𝑞||𝜓𝑘(2)(||𝑤)𝑘=12𝑘𝛽𝑞2𝑘𝐿2𝑘+|𝑤|𝑚+𝐿1|𝑤|𝑚+𝛽𝑞,𝑗=12𝑗𝛼𝑞||𝜓𝑗(1)||1(𝑢,𝑣)||(||𝑢,𝑣)𝑛+𝑚+𝛼𝑞.(4.5)

This inequality together with the trivial inequality 𝑆0𝑓𝑝𝑓𝑝 yields 𝑓𝛼,𝛽𝑝,𝑞𝑓𝛼,𝛽𝑝,𝑞,(2).

To prove the converse, by Lemma 2.6, it suffices to show that𝑛+𝑚×𝑚(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓𝑝||(||𝑢,𝑣)𝛼|𝑤|𝛽𝑞𝑑𝑢𝑑𝑣𝑑𝑤||||(𝑢,𝑣)𝑛+𝑚|𝑤|𝑚𝑓𝑞𝛼,𝛽𝑝,𝑞.(4.6)By the Calderón’s identity (2.1), we write(Δ𝔉𝑢,𝑣;𝑤)𝑀Δ𝑓=𝔉𝑢,𝑣;𝑤𝑀𝜑𝑆0𝑓+𝑗=1𝑘=1Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓,(4.7)

where the series on the right hand side converges in 𝐿𝑝(𝑛+𝑚). Thus, by Minkowski’s inequality and Young’s inequality, we conclude thatΔ𝔉𝑢,𝑣;𝑤𝑀𝑓𝑝Δ𝔉𝑢,𝑣;𝑤𝑀𝜑1𝑆0𝑓𝑝+𝑗=1𝑘=1Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘1𝜓𝑗,𝑘𝑓𝑝.(4.8)

It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of𝐼𝐼𝐼1𝑗𝑘2𝑗𝛼𝑞2𝑘𝛽𝑞Δ𝔉𝑢,𝑣;𝑤𝑀𝜑𝑞1𝑆0𝑓𝑞𝑝,𝐼𝐼𝐼2𝑗𝑘2𝑗𝛼𝑞2𝑘𝛽𝑞𝑗=1𝑘=1(Δ𝔉𝑢,𝑣;𝑤)𝑀𝜓𝑗,𝑘1𝜓𝑗,𝑘𝑓𝑝𝑞.(4.9)

For 𝐼𝐼𝐼1, we have𝐼𝐼𝐼1𝑗𝑘>0>02𝑗𝑞(𝑀𝛼)2𝑘𝑞(𝑀𝛽)+𝑘𝑗>002𝑗𝑞(𝑀𝛼)2𝑘𝑞𝛽+𝑗0𝑘>02𝑗𝑞𝛼2𝑘𝑞(𝑀𝛽)+𝑗𝑘002𝑗𝑞𝛼2𝑘𝑞𝛼𝑆0𝑓𝑞𝑝𝑆0𝑓𝑞𝑝,(4.10)

where we have used the estimateΔ𝔉𝑢,𝑣;𝑤𝑀𝜑12𝑀min{0,𝑗,𝑘,𝑗𝑘}.(4.11)

To estimate 𝐼𝐼𝐼2, by the estimateΔ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘12𝑀min{0,𝑗𝑗,𝑘𝑘,𝑗𝑗+𝑘𝑘},(4.12)

and Hölder’s inequality, we see that 𝐼𝐼𝐼2 is majorized by𝑗,𝑘0<𝑗𝑗0<𝑘𝑘2𝑞[𝑀𝜀(𝛼𝛽)](𝑗𝑗+𝑘𝑘)+0<𝑗𝑗𝑘>(𝑘0)2𝑞[(𝑀𝛼𝜀)(𝑗𝑗)+(𝑘𝑘)(𝛽𝜀)]+𝑗>(𝑗0)0<𝑘𝑘2𝑞(𝑀𝛽𝜀)(𝑘𝑘)2𝑞(𝑗𝑗)(𝛼𝜀)+𝑗>(𝑗0)𝑘>(𝑘0)2𝑞(𝑗𝑗)(𝛼𝜀)2𝑞(𝑘𝑘)(𝛽𝜀)×2𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝑝𝑗>0𝑘>02𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑗,𝑘𝑓𝑞𝑝,(4.13)

proving (4.6), where 𝜀 is a positive number such that 0<𝜀<(𝛼𝛽)<(𝛼𝛽)+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 𝜓𝑗,𝑘, we have𝑗,𝑘+2𝑗𝛼𝑞2𝑘𝛽𝑞||𝜓𝑗,𝑘||𝑓(𝑥,𝑦)𝑞𝑗,𝑘+2𝑗𝛼𝑞2𝑘𝛽𝑞𝑛+𝑚×𝑚||𝜓𝑗(1)||||(𝑢,𝑣)𝜓𝑘(2)|||||(𝑤)(Δ𝔉𝑢,𝑣;𝑤)𝑀|||𝑓(𝑥,𝑦)𝑞𝑑𝑢𝑑𝑣𝑑𝑤𝑛+𝑚×𝑚𝑗=12𝑗𝛼𝑞||𝜓𝑗(1)||(𝑢,𝑣)𝑘=12𝑘𝛽𝑞||𝜓𝑘(2)|||||Δ(𝑤)𝔉𝑢,𝑣;𝑤𝑀|||𝑓(𝑥,𝑦)𝑞𝑑𝑢𝑑𝑣𝑑𝑤𝑛+𝑚×𝑚|||Δ𝔉𝑢,𝑣;𝑤𝑀|||𝑓(𝑥,𝑦)|(𝑢,𝑣)|𝛼|𝑤|𝛽𝑞𝑑𝑢𝑑𝑣𝑑𝑤||(||𝑢,𝑣)𝑛+𝑚|𝑤|𝑚,(4.14)

where ̃𝑔(𝑥)=𝑔(𝑥). Therefore,𝑗=1𝑘=12𝑗𝛼𝑞2𝑘𝛽𝑞||𝜓𝑗,𝑘||𝑓𝑞1/𝑞𝑝𝑓𝛼,𝛽𝑝,𝑞,(2),(4.15)

which, together with the obvious inequality 𝑆0𝑓𝑝𝑓𝑝, yields 𝑓𝛼,𝛽𝑝,𝑞𝑓𝛼,𝛽𝑝,𝑞,(2).

To show the converse, writeΔ𝔉𝑢,𝑣;𝑤𝑀Δ𝑓=𝔉𝑢,𝑣;𝑤𝑀𝜑S0𝑓+𝑗=1𝑘=1Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘𝜓𝑗,𝑘𝑓,(4.16)

where the series converges in 𝒮𝔉(𝑛×𝑚). It follows that𝑛+𝑚×𝑚|(Δ𝔉𝑢,𝑣;𝑤)𝑀𝑓||(𝑢,𝑣)|𝛼|𝑤|𝛽𝑞𝑑𝑢𝑑𝑣𝑑𝑤|(𝑢,𝑣)|𝑛+𝑚|𝑤|𝑚1/𝑞𝑝𝑗𝑘2𝑗𝛼𝑞2𝑘𝛽𝑞|((Δ𝔉𝑢,𝑣;𝑤)𝑀𝜑)𝑆0𝑓|𝑞1/𝑞𝑝+𝑗𝑘2𝑗𝛼𝑞2𝑘𝛽𝑞𝑗=1𝑘=1||||Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘𝜓𝑗,𝑘||||𝑓𝑞1/𝑞𝑝=𝐼𝑉1+𝐼𝑉2,(4.17)where |(𝑢,𝑣)|2𝑗 and |𝑤|2𝑘.

For 𝐼𝑉2, by Lemma 2.2,|||Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘|||(𝑢,𝑣)2𝑀min{0,𝑗𝑗,𝑘𝑘,𝑗𝑗+𝑘𝑘}2𝑗𝑛2(𝑗𝑘)𝑚1+2𝑗|𝑢|+2𝑗𝑘|𝑣|𝐿2.(4.18)

Consequently,|||Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘𝜓𝑗,𝑘|||𝑓(𝑥,𝑦)2𝑀min{0,𝑗𝑗,𝑘𝑘,𝑗𝑗+𝑘𝑘}𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦).(4.19)Therefore,𝑗𝑘2𝑗𝛼𝑞2𝑘𝛽𝑞𝑗=1𝑘=1||||Δ𝔉𝑢,𝑣;𝑤𝑀𝜓𝑗,𝑘𝜓𝑗,𝑘||||𝑓(𝑥,𝑦)𝑞𝑗,𝑘+𝑗𝑘>𝑗>𝑘2𝑞[𝑀𝜀(𝛼𝛽)](𝑗𝑗+𝑘𝑘)+𝑗𝑘>𝑗<𝑘2𝑞(𝑀𝛼𝜀)(𝑗𝑗)2𝑞(𝑘𝑘)(𝛽𝜀)+𝑗𝑘<𝑗>𝑘2𝑞(𝑀𝛽𝜀)(𝑘𝑘)2𝑞(𝑗𝑗)(𝛼𝜀)+𝑗𝑘<𝑗<𝑘2𝑞(𝑗𝑗)(𝛼𝜀)2𝑞(𝑘𝑘)(𝛽𝜀)×2𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑞𝑗,𝑘+2𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑏,𝑗,𝑘(𝑓)(𝑥,𝑦)𝑞,(4.20)

which together with the maximal function characterization of 𝛼,𝛽𝑝,𝑞 implies𝐼𝑉2𝑗,𝑘+2𝑗𝛼𝑞2𝑘𝛽𝑞𝜓𝑏,𝑗,𝑘(𝑓)𝑞1/𝑞𝑝𝑓𝛼,𝛽𝑝,𝑞.(4.21)

As for 𝐼𝑉1, similar estimate to (4.19) yields|||Δ𝔉𝑢,𝑣;𝑤𝑀𝜑𝑆0|||𝑓(𝑥,𝑦)2𝑀max{0,𝑗,𝑘,𝑗+𝑘}𝜓𝑏,0,0(𝑓)(𝑥,𝑦).(4.22)

Therefore,𝐼𝑉1𝑗>0𝑘>02𝑗𝑞(𝑀𝛼)2𝑘𝑞(𝑀𝛽)+𝑗>0𝑘02𝑗𝑞(𝑀𝛼)2𝑘𝑞𝛽+𝑗𝑘0>02𝑗𝑞𝛼2𝑘𝑞(𝑀𝛽)+𝑗𝑘002𝑗𝑞𝛼2𝑘𝑞𝛼1/𝑞𝜓𝑏,0,0(𝑓)𝑝𝑆0𝑓𝑝.(4.23)

This estimate together with (4.21) and Lemma 2.6 yields 𝑓𝛼,𝛽𝑝,𝑞,(2)𝑓𝛼,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.

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