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

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 bcomplex on certain CR submanifolds of C, in 2001, Nagel et al. 2 studied a class of product singular integrals with flag kernel. They proved, among other things, the L 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 developedHardy spacesH F Rn×Rm with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calderón’s identity and proved theH F R n×Rm → H F Rn×Rm andH p F R n×Rm → L Rn×


Introduction and Main Results
The flag singular integral operators were first introduced by M üller, Ricci, and Stein when they studied the Marcinkiewicz multiplier on the Heisenberg groups in 1 .To study the bcomplex on certain CR submanifolds of C n , in 2001, Nagel et  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

1.2
Let x, y 2 jn ψ 1 2 j x, 2 j y , ψ 2 k y 2 km ψ 2 2 k y , and , the following Calder ón's reproducing formula holds: where the series converges in and the seminorm of f is defined by : for all representations of f in 1.7 , 1.7 where , by taking the Fourier transform, where the series converges in L 2 R n × R m norm.Now, we introduce the definition of inhomogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals.
and the inhomogeneous Besov space associated with flag singular integrals 1.11 Throughout this paper, we always work on R n × R m for some fixed n, m and use p,q , and so forth.We would like to point out that the multiparameter structures are involved in the definitions of B α,β p,q and F α,β p,q .The following result shows that the definition of the Besov spaces B α,β p,q and Triebel-Lizorkin spaces F α,β p,q is independent of the choice of ψ 1 , ψ 2 , ϕ ; thus, the Besov spaces B α,β p,q and the Triebel-Lizorkin spaces F α,β p,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 , 1.12 Remark 1.4.As the classical case, it is not hard to show that are norms of B α,β p,q and F α,β p,q , respectively.Moreover, B α,β p,q and F α,β p,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 .

1.13
For j k 0, define 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.
Here and in what follows, one uses the following notation: 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, for all multi-index α and every normalized bump function φ 1 on R m and every δ > 0, for every multi-index β and every normalized bump function φ 2 on R n and every δ > 0, for every normalized bump function φ 3 on R m and every δ 1 > 0 and δ 2 > 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.
As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will give the difference characterization for B α,β p,q and F α,β p,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 , where • denotes the greatest integer function, one defines 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.

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

Lemma 2.1. The inhomogeneous Calderón's reproducing formula holds
where the series converges in We point out that in 8 the homogeneous Calder ón'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 We need to show that for all N ∈ Z , 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, On the other hand, using the cancellation conditions of ψ , and the smoothness of f # , we can get Now, 2.5 together with the estimate 2.6 implies that, for Applying this to ∂ u f # here u denotes any multi-index in N n 2m and noting that 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 .

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 where ψ, ϕ are defined as in Section 1.

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 b b 1 , b 2 ∈ R ×R , there exists a constant C C L, b depending only on L and b, but independent on j, k, j , k , such that

2.10
Proof.By the almost orthogonality estimate in Lemma 2.2, for any L > 0 and M > b 1 ∨ b 2 , we have the pointwise estimate
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 by 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 < c 1 , c 2 < ∞, and 0 < r < ∞, then for all j, k ∈ Z and for all C 1 functions u on R n × R m whose Fourier transform is supported in the rectangle {ξ

2.14
Lemma 2.5 can be proved as in the classical one-parameter case.We refer the reader to 11 .

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:

2.15
Proof.For f ∈ S F R n × R m , by inhomogeneous Calder ón's reproducing formula 2.1 , where we have used H ölder's inequality in the last inequality.This proves Lemma 2.6 for Besov spaces.

Proof of Theorems 1.3 and 1.5
We first prove Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces by showing 3.1 The first inequality in 3.1 follows from the pointwise inequality Next, for any admissible b, fix b.Since b is admissible, we can choose r ≤ p ∧ q such that b 1 ≥ n/r, b 2 ≥ m/r, and the thus inequality 2.14 holds.We apply Lemma 2.5 and the L p/r q/r boundedness of M s to deduce 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 Calder ón's identity 2.1 , Minkowski's inequality, Remark 2.4, and Lemma 2.5, we deduce that

3.4
To finish the proof, it remains to show

3.5
By the inhomogeneous Calder ón's identity 2.1 , we have It follows that

3.7
We first estimate I 1 .By the support properties of ψ j,k and ϕ and Young's inequality, Next, we give the estimate for where we have used Lemma 2.2 in the last inequality.Applying Minkowski's inequality and H ölder's inequality yields where we have chosen ε as a small positive constant less than L − |α| ∨ |β| .Therefore,

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 Calder ón'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 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.

Proof of Theorem 1.8 for Besov Space.
By the moment conditions of ψ 1 j 's and ψ 2 k 's, we may write

Journal of Function Spaces and Applications
Hence, where the last inequality follows from

4.5
This inequality together with the trivial inequality S 0 f p f p yields f B α,β p,q f B α,β p,q, 2 .
To prove the converse, by Lemma 2.6, it suffices to show that By the Calder ón's identity 2.1 , we write 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 It follows that the left hand side in 4.6 is dominated, up to a constant, by the sum of 4.9 For III 1 , we have where we have used the estimate To estimate III 2 , by the estimate 12 and H ölder's inequality, we see that III 2 is majorized by 2 jαq 2 kβq ψ j,k * f q p , 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.

Proof of Theorem 1.8 for Triebel-Lizorkin Space.
Using the moment conditions on ψ j,k , we have j,k∈Z

4.23
This estimate together with 4.21 and Lemma 2.6 yields f F α,β p,q, 2 f F α,β p,q . This ends the proof of Theorem 1.8.

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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 p R n ×R m with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calder ón's identity and proved the H p R n ×R m → H p R n ×R m and H p R n ×R m → L p R n × R m boundedness for flag singular integral operators.The duality of H p 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 .

j:jn 2 j∧k m 1 2 j |u| 2 j∧k |v| L 2 .
with the obvious inequality S 0 f p f p , yields f F α,β k * ψ j,k * f,4.16where the series converges inS F R n × R m .It follows that R n m ×R m | Δ F u,v;w M f| | u, v | α |w| β q du dv dw | u, v | n m |w| m ∈Z k ∈Z 2 j αq 2 k βq | Δ F u,v;w M ϕ * S 0 f| q } IV 1 IV 2 , 4.17 where | u, v | ∼ 2 −j and |w| ∼ 2 −k .For IV 2 , by Lemma 2.2,Δ F u,v;w M ψ j,k u, v 2 M•min{0,j−j ,k−k ,j−j k−k } 2 k * ψ j,k * f x, y 2 M•min{0,j−j ,k−k ,j−j k−k } ψ * k * ψ j,k * f x, y for IV 1 , similar estimate to 4.19 yieldsΔ F u,v;w M ϕ * S 0 f x, y 2 −M•max{0,j ,k ,j k } ψ * b,0,0 f x, y .4.22 Journal of Function Spaces and Applicationswhere Δ u,v is the difference operator on R m n , and Δ 2 w is the difference operator on R m .For k ∈ Z and k ≥ 2, the kth flag difference operator Δ F