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

Authors: Xinfeng Wu; Zongguang Liu
Journal: Journal of Function Spaces and Applications
 (2012)

---

## 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.


---

## Body

## 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].For0
<
p
≤
1, Han and Lu [7] developed Hardy spaces H
𝔉
p
(
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
𝔉
p
(
ℝ
n
×
ℝ
m
)
→
H
𝔉
p
(
ℝ
n
×
ℝ
m
) and H
𝔉
p
(
ℝ
n
×
ℝ
m
)
→
L
p
(
ℝ
n
×
ℝ
m
) boundedness for flag singular integral operators. The duality of H
𝔉
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].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 haveflag 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. Letψ
(
1
)
∈
𝒮
(
ℝ
n
×
ℝ
m
) with(1.1)
supp
⁡
ψ
(
1
)
̂
⊂
{
(
ξ
,
η
)
∈
R
n
×
R
m
:
1
2
≤
|
ξ
,
η
|
≤
2
}
,
∑
j
∈
Z
|
ψ
(
1
)
̂
(
2
-
j
ξ
,
2
-
j
η
)
|
2
=
1
∀
(
ξ
,
η
)
∈
R
n
×
R
m
∖
{
(
0,0
)
}
,and letψ
(
2
)
∈
𝒮
(
ℝ
m
) with(1.2)
supp
⁡
ψ
(
2
)
̂
⊂
{
η
∈
R
m
:
1
2
≤
|
η
|
≤
2
}
,
∑
k
∈
Z
|
ψ
(
2
)
̂
(
2
-
k
η
)
|
2
=
1
∀
η
∈
R
m
∖
{
0
}
.Letψ
j
(
1
)
(
x
,
y
)
=
2
j
n
ψ
(
1
)
(
2
j
x
,
2
j
y
)
,
ψ
k
(
2
)
(
y
)
=
2
k
m
ψ
(
2
)
(
2
k
y
), and(1.3)
ψ
j
,
k
(
x
,
y
)
=
ψ
j
(
1
)
*
2
ψ
k
(
2
)
(
x
,
y
)
=
∫
ψ
j
(
1
)
(
x
,
y
-
z
)
ψ
k
(
2
)
(
z
)
d
z
,then, forf
∈
L
2
(
ℝ
n
×
ℝ
m
), the following Calderón’s reproducing formula holds:(1.4)
f
=
∑
j
∈
Z
∑
k
∈
Z
ψ
j
,
k
*
ψ
j
,
k
*
f
,where the series converges inL
2
(
ℝ
n
×
ℝ
m
).A Schwartz functionf
∈
𝒮
(
ℝ
n
+
2
m
) is said to be a product test function in 𝒮
∞
(
ℝ
n
+
m
×
ℝ
m
) if it satisfies(1.5)
∫
R
m
×
R
n
f
(
x
,
y
,
z
)
x
α
y
β
d
x
d
y
=
∫
R
m
f
(
x
,
y
,
z
)
z
γ
d
z
=
0
,for all multi-indicesα
∈
ℕ
n
,
β
,
γ
∈
ℕ
m.Definition 1.1.
A functionf
(
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
(1.6)
f
(
x
,
y
)
=
∫
R
m
f
#
(
x
,
y
-
z
,
z
)
d
z
,and the seminorm of f is defined by
(1.7)
‖
f
‖
F
,
α
,
β
=
inf
⁡
{
‖
f
#
‖
α
,
β
:
for
all
representations
of
f
in
(
1.7
)
}
,
where ∥
·
∥
α
,
β denote the seminorm in 𝒮
(
ℝ
n
+
2
m
). Denote by 𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
) the dual of 𝒮
𝔉
(
ℝ
n
×
ℝ
m
).Choose a Schwartz functionφ on ℝ
n
×
ℝ
m such that(1.8)
|
φ
̂
(
ξ
,
η
)
|
2
=
1
-
∑
j
=
1
∞
∑
k
=
1
∞
|
ψ
(
1
)
̂
(
2
-
j
ξ
1
,
2
-
j
ξ
2
)
|
2
|
ψ
(
2
)
̂
(
2
-
k
ξ
2
)
|
2
.Note that φ
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
) with Fourier transform is supported in {
(
ξ
,
η
)
:
|
(
ξ
,
η
)
|
≤
1
}. Define the operator S
0 by S
0
f
=
φ
*
f
,
f
∈
𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
).Forf
∈
L
2
(
ℝ
n
×
ℝ
m
), by taking the Fourier transform,(1.9)
f
=
φ
*
φ
*
f
+
∑
j
=
1
∞
∑
k
=
1
∞
ψ
j
,
k
*
ψ
j
,
k
*
f
,where the series converges inL
2
(
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.
Letα
,
β
∈
(
-
∞
,
∞
) and p
,
q
∈
(
1
,
∞
). The inhomogeneous Triebel-Lizorkin space associated with flag singular integrals
ℱ
p
,
q
α
,
β
(
ℝ
n
×
ℝ
m
) is defined to be the collection of all f
∈
𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
) such that
(1.10)
‖
f
‖
F
p
,
q
α
,
β
(
R
n
×
R
m
)
ψ
:
=
‖
S
0
f
‖
L
p
(
R
n
×
R
m
)
+
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
|
q
)
1
/
q
‖
L
p
(
R
n
×
R
m
)
<
∞
,
and the inhomogeneous Besov space associated with flag singular integrals
ℬ
p
,
q
α
,
β
(
ℝ
n
×
ℝ
m
) is defined to be the collection of all f
∈
𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
) such that
(1.11)
‖
f
‖
B
p
,
q
α
,
β
(
R
n
×
R
m
)
ψ
:
=
‖
S
0
f
‖
L
p
(
R
n
×
R
m
)
+
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
L
p
(
R
n
×
R
m
)
q
)
1
/
q
<
∞
.Throughout this paper, we always work onℝ
n
×
ℝ
m for some fixed n
,
m and use ℱ
p
,
q
α
,
β to denote ℱ
p
,
q
α
,
β
(
ℝ
n
×
ℝ
m
), similarly for 𝒮
𝔉
,
ℬ
p
,
q
α
,
β, and so forth. We would like to point out that the multiparameter structures are involved in the definitions of ℬ
p
,
q
α
,
β and ℱ
p
,
q
α
,
β. The following result shows that the definition of the Besov spaces ℬ
p
,
q
α
,
β and Triebel-Lizorkin spaces ℱ
p
,
q
α
,
β is independent of the choice of (
ψ
(
1
)
,
ψ
(
2
)
,
φ
); thus, the Besov spaces ℬ
p
,
q
α
,
β and the Triebel-Lizorkin spaces ℱ
p
,
q
α
,
β are well defined.Theorem 1.3.
Ifθ
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)
‖
f
‖
F
p
,
q
α
,
β
θ
~
‖
f
‖
F
p
,
q
α
,
β
ψ
,
‖
f
‖
B
p
,
q
α
,
β
θ
~
‖
f
‖
B
p
,
q
α
,
β
ψ
.Remark 1.4.
As the classical case, it is not hard to show that∥
·
∥
ℬ
p
,
q
α
,
β
ψ and ∥
·
∥
ℱ
p
,
q
α
,
β
ψ are norms of ℬ
p
,
q
α
,
β and ℱ
p
,
q
α
,
β, respectively. Moreover, ℬ
p
,
q
α
,
β and ℱ
p
,
q
α
,
β are complete with respect to these norms and hence are Banach spaces. We omit the details.Throughout this paper, we use the notationsj
∧
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 b
⃗
=
(
b
1
,
b
2
)
∈
ℝ
+
×
ℝ
+, define(1.13)
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
=
sup
⁡
(
u
,
v
)
∈
R
n
×
R
m
|
ψ
j
,
k
*
f
(
x
-
u
,
y
-
v
)
|
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
.Forj
=
k
=
0, define(1.14)
φ
b
⃗
,
0,0
*
(
f
)
(
x
)
=
sup
⁡
y
∈
R
n
|
φ
*
f
(
x
-
u
,
y
-
v
)
|
(
1
+
|
u
|
)
b
1
(
1
+
|
v
|
)
b
2
.An indexb
⃗
=
(
b
1
,
b
2
) is said to be admissible if(1.15)
b
1
>
n
p
∧
q
,
b
2
>
m
p
∧
q
.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.
Letα
,
β
∈
ℝ and p
,
q
∈
(
1
,
∞
). If b
⃗ is admissible, then for f
∈
𝒮
𝔉
′, one has(i)
∥
f
∥
ℬ
p
,
q
,
(
1
)
α
,
β
:
=
∥
S
0
f
∥
p
+
∥
{
2
α
j
2
β
k
∥
ψ
b
⃗
,
j
,
k
*
(
f
)
∥
p
}
j
,
k
∥
ℓ
q
~
∥
f
∥
ℬ
p
,
q
α
,
β,
(ii)
∥
f
∥
ℱ
p
,
q
,
(
1
)
α
,
β
:
=
∥
S
0
f
∥
p
+
∥
∥
{
2
α
j
2
β
k
ψ
b
⃗
,
j
,
k
*
(
f
)
}
j
,
k
∥
ℓ
q
∥
L
p
~
∥
f
∥
ℱ
p
,
q
α
,
β. Here and in what follows, one uses the following notation:
(1.16)
‖
{
a
j
,
k
}
j
,
k
‖
l
q
:
=
{
∑
j
=
1
∞
∑
k
=
1
∞
|
a
j
,
k
|
q
}
1
/
q
.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 onℝ
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-indicesα and  β,
(1.17)
|
∂
x
α
∂
y
β
K
(
x
,
y
)
|
≤
C
α
,
β
|
x
|
-
n
-
|
α
|
(
|
x
|
+
|
y
|
)
-
m
-
|
β
|
,
for all (
x
,
y
)
∈
ℝ
n
×
ℝ
m with |
x
|
≠
0,
(2)
(Cancellation condition)(1.18)
|
∫
R
m
∂
x
α
K
(
x
,
y
)
ϕ
1
(
δ
y
)
d
y
|
≤
C
α
|
x
|
-
n
-
|
α
|
,
for all multi-index α and every normalized bump function ϕ
1 on ℝ
m and every δ
>
0,
(1.19)
|
∫
R
n
∂
y
β
K
(
x
,
y
)
ϕ
2
(
δ
x
)
d
x
|
≤
C
β
|
y
|
-
m
-
|
β
|
for every multi-index β and every normalized bump function ϕ
2 on ℝ
n and every δ
>
0,
(1.20)
|
∫
R
n
+
m
K
(
x
,
y
)
ϕ
3
(
δ
1
x
,
δ
2
y
)
d
x
d
y
|
≤
C
,
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.Theorem 1.7.
Suppose thatT 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
p
,
q
α
,
β and on ℬ
p
,
q
α
,
β.As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will give the difference characterization forℬ
p
,
q
α
,
β and ℱ
p
,
q
α
,
β. 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 by(1.21)
Δ
u
,
v
;
w
F
f
(
x
,
y
)
=
[
Δ
u
,
v
∘
Δ
w
(
2
)
]
f
(
x
,
y
)
=
f
(
x
+
u
,
y
+
v
+
w
)
-
f
(
x
+
u
,
y
+
v
)
-
f
(
x
,
y
+
w
)
+
f
(
x
,
y
)
,whereΔ
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 k
th flag difference operator
(
Δ
u
,
v
;
w
𝔉
)
k can be defined inductively by(1.22)
(
Δ
u
,
v
;
w
F
)
k
=
Δ
u
,
v
;
w
F
∘
(
Δ
u
,
v
;
w
F
)
k
-
1
.Theorem 1.8.
Ifα
,
β
>
0, 1
<
p
<
∞, 1
<
q
<
∞, and M
≥
[
α
∨
β
]
+
1, where [
·
] denotes the greatest integer function, one defines
(1.23)
‖
f
‖
B
p
,
q
,
(
2
)
α
,
β
:
=
‖
f
‖
p
+
{
∫
R
n
+
m
×
R
m
(
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
}
1
/
q
,
‖
f
‖
F
p
,
q
,
(
2
)
α
,
β
:
=
‖
f
‖
p
+
‖
{
∫
R
n
+
m
×
R
m
(
|
(
Δ
u
,
v
;
w
F
)
M
f
|
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
}
1
/
q
‖
p
,
then ∥
f
∥
ℬ
p
,
q
,
(
2
)
α
,
β
~
∥
f
∥
ℬ
p
,
q
α
,
β
,
∥
f
∥
ℱ
p
,
q
,
(
2
)
α
,
β
~
∥
f
∥
ℱ
p
,
q
α
,
β.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 Section2, 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 in𝒮
𝔉
′

Lemma 2.1.
The inhomogeneous Calderón’s reproducing formula holds(2.1)
f
=
φ
*
φ
*
f
+
∑
j
=
1
∞
∑
k
=
1
∞
ψ
j
,
k
*
ψ
j
,
k
*
f
,
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 anyf
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
), then by definition, there exists f
#
∈
𝒮
(
ℝ
n
+
m
×
ℝ
m
) such that f
(
x
,
y
)
=
∫
ℝ
m
f
#
(
x
,
y
-
z
,
z
)
d
z. We need to show that for all N
∈
ℤ
+,
(2.2)
φ
#
*
φ
#
*
f
#
+
∑
j
,
k
∈
Z
+
:
j
≤
N
or
k
≤
N
ψ
j
,
k
#
*
ψ
j
,
k
#
*
f
#
,
tends to f in the topology of 𝒮
(
ℝ
n
+
2
m
) 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,
(2.3)
f
#
(
x
,
y
)
-
f
N
#
(
x
,
y
)
=
(
f
̂
#
-
f
̂
N
#
)
∨
(
x
,
y
)
,
for
every
(
x
,
y
)
∈
R
n
+
m
×
R
m
.
Let h
N
(
ξ
,
η
,
x
,
y
)
=
e
i
(
x
·
ξ
+
y
·
η
)
f
̂
#
(
ξ
,
η
)
∑
k
=
N
+
1
∞
|
ψ
(
2
)
̂
(
2
-
k
η
)
|
2 so that
(2.4)
f
#
(
x
,
y
)
-
f
N
#
(
x
,
y
)
=
c
n
∫
R
n
+
m
×
R
m
h
N
(
ξ
,
η
,
x
,
y
)
d
η
d
ξ
.
Since |
h
N
(
ξ
,
η
,
x
,
y
)
|
≤
|
f
̂
#
(
ξ
,
η
)
|
∈
L
1
(
ℝ
n
+
m
×
ℝ
m
), Lebesgue’s dominated convergence theorem yields
(2.5)
lim
⁡
N
→
+
∞
f
N
#
(
x
,
y
)
=
f
#
(
x
,
y
)
,
for
every
(
x
,
y
)
∈
R
n
+
m
×
R
m
.
On the other hand, using the cancellation conditions ofψ
j
(
1
)
*
ψ
j
(
1
)
,
ψ
k
(
2
)
*
ψ
k
(
2
), and the smoothness of f
#, we can get
(2.6)
|
ψ
j
,
k
#
*
ψ
j
,
k
#
*
f
#
(
x
,
y
)
|
≲
2
-
j
2
-
k
(
1
+
|
x
|
+
|
y
|
)
-
L
,
∀
j
,
k
,
L
∈
Z
+
.
Now, (2.5) together with the estimate (2.6) implies that, for f
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
),
(2.7)
|
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
.
Applying this to ∂
u
f
# (here u denotes any multi-index in ℕ
n
+
2
m) and noting that ∂
u
f
N
#
=
(
∂
u
f
#
)
N, we obtain
(2.8)
lim
⁡
N
→
+
∞
sup
⁡
(
x
,
y
)
∈
R
n
×
R
m
(
1
+
|
x
|
+
|
y
|
)
L
|
∂
u
(
f
#
-
f
N
#
)
(
x
,
y
)
|
=
0
.
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.
Letx
∈
ℝ
n
,
y
∈
ℝ
m. Given any positive integers L and M, there exists a constant C
=
C
(
L
,
M
)
>
0 such that
(2.9)
|
ψ
j
,
k
*
φ
j
′
,
k
′
(
x
,
y
)
|
≤
C
2
-
|
j
-
j
′
|
L
2
-
|
k
-
k
′
|
L
2
(
j
∧
j
′
)
M
(
2
j
∧
j
′
+
|
x
|
)
n
+
M
2
(
j
∧
j
′
∧
k
∧
k
′
)
M
(
2
j
∧
j
′
∧
k
∧
k
′
+
|
y
|
)
m
+
M
,
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.
Forj
,
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)
|
(
ψ
j
,
k
*
ψ
j
′
,
k
′
*
ψ
j
′
,
k
′
*
f
)
(
x
,
y
)
|
≤
C
2
-
L
|
j
-
j
′
|
2
-
L
|
k
-
k
′
|
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
.Proof.
By the almost orthogonality estimate in Lemma2.2, for any L
>
0 and M
>
b
1
∨
b
2, we have the pointwise estimate
(2.11)
|
(
ψ
j
,
k
*
ψ
j
′
,
k
′
*
ψ
j
′
,
k
′
*
f
)
(
x
,
y
)
|
≲
2
-
|
j
-
j
′
|
L
′
2
-
|
k
-
k
′
|
L
′
×
∫
R
n
×
R
m
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
d
u
d
v
≤
2
-
|
j
-
j
′
|
L
′
2
-
|
k
-
k
′
|
L
′
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
)
×
∫
R
n
×
R
m
2
(
j
∧
j
′
)
n
2
(
j
∧
k
∧
j
′
∧
k
′
)
m
(
1
+
2
j
′
|
u
|
)
b
1
(
1
+
2
j
′
∧
k
′
|
v
|
)
b
2
(
1
+
2
j
∧
j
′
|
u
|
)
M
+
n
(
1
+
2
j
∧
k
∧
j
′
∧
k
′
|
v
|
)
M
+
m
d
u
d
v
≲
2
-
|
j
-
j
′
|
(
L
′
-
b
1
)
2
-
|
k
-
k
′
|
(
L
′
-
b
1
-
b
2
)
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
×
∫
R
n
×
R
m
2
(
j
∧
j
′
)
n
2
(
j
∧
k
∧
j
′
∧
k
′
)
m
(
1
+
2
j
∧
j
′
|
u
|
)
M
+
n
-
b
1
(
1
+
2
j
∧
k
∧
j
′
∧
k
′
|
v
|
)
M
+
m
-
b
2
d
u
d
v
≲
2
-
|
j
-
j
′
|
(
L
)
2
-
|
k
-
k
′
|
(
L
)
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
,
where L
=
L
′
-
b
1
-
b
2. This proves (2.10).Remark 2.4.
Since the almost orthogonality estimates hold withψ
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 byℳ
s the strong maximal operator defined by(2.12)
M
s
(
f
)
(
x
)
=
sup
⁡
R
∋
x
∫
R
|
f
(
y
)
|
d
y
,where the supremum is taken over all open rectanglesR in ℝ
n
×
ℝ
m that contain the point x.Lemma 2.5.
Let0
<
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
:
|
ξ
′
|
≤
c
1
2
j
,
|
ξ
n
|
≤
c
2
2
j
∧
k
}, one has the estimate
(2.13)
sup
⁡
y
∈
R
n
|
u
(
x
-
u
,
y
-
v
)
|
(
1
+
2
j
|
u
|
)
n
/
r
(
1
+
2
j
∧
k
|
v
|
)
m
/
r
≤
C
M
s
(
|
u
|
r
)
(
x
,
y
)
1
/
r
.
In particular, if b
⃗
=
(
b
1
,
b
2
) with b
1
≥
n
/
r and b
2
≥
m
/
r, then for all f
∈
L
p
(
ℝ
n
×
ℝ
m
)
(
1
<
p
<
∞
),
(2.14)
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
≲
M
s
(
|
ψ
j
,
k
*
f
|
r
)
(
x
,
y
)
1
/
r
.Lemma2.5 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.
Forα
,
β
>
0 and p
,
q
∈
(
1
,
∞
), one has the following continuous embedding:
(2.15)
B
p
,
q
α
,
β
↪
L
p
,
F
p
,
q
α
,
β
↪
L
p
.Proof.
Forf
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
), by inhomogeneous Calderón’s reproducing formula (2.1),
(2.16)
‖
f
‖
p
≲
‖
S
0
f
‖
p
+
∑
j
=
1
∞
∑
k
=
1
∞
‖
ψ
j
,
k
*
f
‖
p
≲
‖
S
0
f
‖
p
+
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
)
1
/
q
~
‖
f
‖
B
p
,
q
α
,
β
,
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
(2.17)
‖
f
‖
p
≲
‖
φ
*
(
S
0
f
)
‖
p
+
‖
∑
j
=
1
∞
∑
k
=
1
∞
|
ψ
j
,
k
*
ψ
j
,
k
*
f
|
‖
p
≲
‖
S
0
f
‖
p
+
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
q
)
1
/
q
‖
p
≲
‖
S
0
f
‖
p
+
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
|
q
)
1
/
q
‖
p
~
‖
f
‖
F
p
,
q
α
,
β
.
This ends the proof of Lemma 2.6.


## 2.1. Inhomogeneous Calderón’s Reproducing Formula in𝒮
𝔉
′

Lemma 2.1.
The inhomogeneous Calderón’s reproducing formula holds(2.1)
f
=
φ
*
φ
*
f
+
∑
j
=
1
∞
∑
k
=
1
∞
ψ
j
,
k
*
ψ
j
,
k
*
f
,
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 anyf
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
), then by definition, there exists f
#
∈
𝒮
(
ℝ
n
+
m
×
ℝ
m
) such that f
(
x
,
y
)
=
∫
ℝ
m
f
#
(
x
,
y
-
z
,
z
)
d
z. We need to show that for all N
∈
ℤ
+,
(2.2)
φ
#
*
φ
#
*
f
#
+
∑
j
,
k
∈
Z
+
:
j
≤
N
or
k
≤
N
ψ
j
,
k
#
*
ψ
j
,
k
#
*
f
#
,
tends to f in the topology of 𝒮
(
ℝ
n
+
2
m
) 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,
(2.3)
f
#
(
x
,
y
)
-
f
N
#
(
x
,
y
)
=
(
f
̂
#
-
f
̂
N
#
)
∨
(
x
,
y
)
,
for
every
(
x
,
y
)
∈
R
n
+
m
×
R
m
.
Let h
N
(
ξ
,
η
,
x
,
y
)
=
e
i
(
x
·
ξ
+
y
·
η
)
f
̂
#
(
ξ
,
η
)
∑
k
=
N
+
1
∞
|
ψ
(
2
)
̂
(
2
-
k
η
)
|
2 so that
(2.4)
f
#
(
x
,
y
)
-
f
N
#
(
x
,
y
)
=
c
n
∫
R
n
+
m
×
R
m
h
N
(
ξ
,
η
,
x
,
y
)
d
η
d
ξ
.
Since |
h
N
(
ξ
,
η
,
x
,
y
)
|
≤
|
f
̂
#
(
ξ
,
η
)
|
∈
L
1
(
ℝ
n
+
m
×
ℝ
m
), Lebesgue’s dominated convergence theorem yields
(2.5)
lim
⁡
N
→
+
∞
f
N
#
(
x
,
y
)
=
f
#
(
x
,
y
)
,
for
every
(
x
,
y
)
∈
R
n
+
m
×
R
m
.
On the other hand, using the cancellation conditions ofψ
j
(
1
)
*
ψ
j
(
1
)
,
ψ
k
(
2
)
*
ψ
k
(
2
), and the smoothness of f
#, we can get
(2.6)
|
ψ
j
,
k
#
*
ψ
j
,
k
#
*
f
#
(
x
,
y
)
|
≲
2
-
j
2
-
k
(
1
+
|
x
|
+
|
y
|
)
-
L
,
∀
j
,
k
,
L
∈
Z
+
.
Now, (2.5) together with the estimate (2.6) implies that, for f
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
),
(2.7)
|
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
.
Applying this to ∂
u
f
# (here u denotes any multi-index in ℕ
n
+
2
m) and noting that ∂
u
f
N
#
=
(
∂
u
f
#
)
N, we obtain
(2.8)
lim
⁡
N
→
+
∞
sup
⁡
(
x
,
y
)
∈
R
n
×
R
m
(
1
+
|
x
|
+
|
y
|
)
L
|
∂
u
(
f
#
-
f
N
#
)
(
x
,
y
)
|
=
0
.
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.
Letx
∈
ℝ
n
,
y
∈
ℝ
m. Given any positive integers L and M, there exists a constant C
=
C
(
L
,
M
)
>
0 such that
(2.9)
|
ψ
j
,
k
*
φ
j
′
,
k
′
(
x
,
y
)
|
≤
C
2
-
|
j
-
j
′
|
L
2
-
|
k
-
k
′
|
L
2
(
j
∧
j
′
)
M
(
2
j
∧
j
′
+
|
x
|
)
n
+
M
2
(
j
∧
j
′
∧
k
∧
k
′
)
M
(
2
j
∧
j
′
∧
k
∧
k
′
+
|
y
|
)
m
+
M
,
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.
Forj
,
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)
|
(
ψ
j
,
k
*
ψ
j
′
,
k
′
*
ψ
j
′
,
k
′
*
f
)
(
x
,
y
)
|
≤
C
2
-
L
|
j
-
j
′
|
2
-
L
|
k
-
k
′
|
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
.Proof.
By the almost orthogonality estimate in Lemma2.2, for any L
>
0 and M
>
b
1
∨
b
2, we have the pointwise estimate
(2.11)
|
(
ψ
j
,
k
*
ψ
j
′
,
k
′
*
ψ
j
′
,
k
′
*
f
)
(
x
,
y
)
|
≲
2
-
|
j
-
j
′
|
L
′
2
-
|
k
-
k
′
|
L
′
×
∫
R
n
×
R
m
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
d
u
d
v
≤
2
-
|
j
-
j
′
|
L
′
2
-
|
k
-
k
′
|
L
′
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
)
×
∫
R
n
×
R
m
2
(
j
∧
j
′
)
n
2
(
j
∧
k
∧
j
′
∧
k
′
)
m
(
1
+
2
j
′
|
u
|
)
b
1
(
1
+
2
j
′
∧
k
′
|
v
|
)
b
2
(
1
+
2
j
∧
j
′
|
u
|
)
M
+
n
(
1
+
2
j
∧
k
∧
j
′
∧
k
′
|
v
|
)
M
+
m
d
u
d
v
≲
2
-
|
j
-
j
′
|
(
L
′
-
b
1
)
2
-
|
k
-
k
′
|
(
L
′
-
b
1
-
b
2
)
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
×
∫
R
n
×
R
m
2
(
j
∧
j
′
)
n
2
(
j
∧
k
∧
j
′
∧
k
′
)
m
(
1
+
2
j
∧
j
′
|
u
|
)
M
+
n
-
b
1
(
1
+
2
j
∧
k
∧
j
′
∧
k
′
|
v
|
)
M
+
m
-
b
2
d
u
d
v
≲
2
-
|
j
-
j
′
|
(
L
)
2
-
|
k
-
k
′
|
(
L
)
ψ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
,
where L
=
L
′
-
b
1
-
b
2. This proves (2.10).Remark 2.4.
Since the almost orthogonality estimates hold withψ
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 byℳ
s the strong maximal operator defined by(2.12)
M
s
(
f
)
(
x
)
=
sup
⁡
R
∋
x
∫
R
|
f
(
y
)
|
d
y
,where the supremum is taken over all open rectanglesR in ℝ
n
×
ℝ
m that contain the point x.Lemma 2.5.
Let0
<
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
:
|
ξ
′
|
≤
c
1
2
j
,
|
ξ
n
|
≤
c
2
2
j
∧
k
}, one has the estimate
(2.13)
sup
⁡
y
∈
R
n
|
u
(
x
-
u
,
y
-
v
)
|
(
1
+
2
j
|
u
|
)
n
/
r
(
1
+
2
j
∧
k
|
v
|
)
m
/
r
≤
C
M
s
(
|
u
|
r
)
(
x
,
y
)
1
/
r
.
In particular, if b
⃗
=
(
b
1
,
b
2
) with b
1
≥
n
/
r and b
2
≥
m
/
r, then for all f
∈
L
p
(
ℝ
n
×
ℝ
m
)
(
1
<
p
<
∞
),
(2.14)
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
≲
M
s
(
|
ψ
j
,
k
*
f
|
r
)
(
x
,
y
)
1
/
r
.Lemma2.5 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.
Forα
,
β
>
0 and p
,
q
∈
(
1
,
∞
), one has the following continuous embedding:
(2.15)
B
p
,
q
α
,
β
↪
L
p
,
F
p
,
q
α
,
β
↪
L
p
.Proof.
Forf
∈
𝒮
𝔉
(
ℝ
n
×
ℝ
m
), by inhomogeneous Calderón’s reproducing formula (2.1),
(2.16)
‖
f
‖
p
≲
‖
S
0
f
‖
p
+
∑
j
=
1
∞
∑
k
=
1
∞
‖
ψ
j
,
k
*
f
‖
p
≲
‖
S
0
f
‖
p
+
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
)
1
/
q
~
‖
f
‖
B
p
,
q
α
,
β
,
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
(2.17)
‖
f
‖
p
≲
‖
φ
*
(
S
0
f
)
‖
p
+
‖
∑
j
=
1
∞
∑
k
=
1
∞
|
ψ
j
,
k
*
ψ
j
,
k
*
f
|
‖
p
≲
‖
S
0
f
‖
p
+
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
q
)
1
/
q
‖
p
≲
‖
S
0
f
‖
p
+
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
|
q
)
1
/
q
‖
p
~
‖
f
‖
F
p
,
q
α
,
β
.
This ends the proof of Lemma 2.6.


## 3. Proof of Theorems1.3 and 1.5

We first prove Theorems1.3 and 1.5 for Triebel-Lizorkin spaces by showing(3.1)
‖
f
‖
F
p
,
q
α
,
β
ψ
≲
‖
f
‖
F
p
,
,
q
,
(
1
)
α
,
β
ψ
≲
‖
f
‖
F
p
,
,
q
,
(
1
)
α
,
β
θ
≲
‖
f
‖
F
p
,
q
α
,
β
θ
.The first inequality in (3.1) follows from the pointwise inequality(3.2)
|
ψ
j
,
k
*
f
(
x
,
y
)
|
≤
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
,
∀
j
,
k
∈
Z
+
.Next, for any admissibleb, 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
(
ℓ
q
/
r
) boundedness of ℳ
s to deduce(3.3)
‖
f
‖
F
p
,
q
,
(
1
)
α
,
β
θ
≲
‖
{
2
j
α
2
k
β
θ
b
⃗
,
j
,
k
*
(
f
)
}
j
,
k
‖
L
p
(
l
q
)
≲
‖
{
2
j
α
2
k
β
M
s
(
|
θ
j
,
k
*
f
|
r
)
1
/
r
}
j
,
k
‖
L
p
(
l
q
)
≲
‖
f
‖
F
p
,
q
α
,
β
θ
,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(3.4)
‖
φ
*
f
‖
p
≲
‖
φ
*
ϕ
‖
1
⋅
‖
ϕ
*
f
‖
p
+
‖
θ
b
⃗
,
1,1
*
(
f
)
‖
p
≲
‖
ϕ
*
f
‖
p
+
‖
{
2
j
α
2
k
β
θ
b
⃗
,
j
,
k
*
(
f
)
}
j
,
k
‖
L
p
(
l
q
)
≲
‖
f
‖
F
p
,
q
,
(
1
)
α
,
β
θ
.To finish the proof, it remains to show(3.5)
‖
{
2
j
α
2
k
β
ψ
b
⃗
,
j
,
k
*
(
f
)
}
j
,
k
‖
L
p
(
l
q
)
≲
‖
f
‖
F
p
,
q
,
(
1
)
α
,
β
θ
.By the inhomogeneous Calderón’s identity (2.1), we have(3.6)
|
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
)
.It follows that(3.7)
‖
{
2
j
α
2
k
β
ψ
b
⃗
,
j
,
k
*
(
f
)
}
j
,
k
‖
L
p
(
l
q
)
≤
‖
{
2
j
α
2
k
β
sup
⁡
(
u
,
v
)
∈
R
n
×
R
m
(
|
ψ
j
,
k
*
ϕ
|
*
|
ϕ
*
f
|
)
(
⋅
-
u
,
⋅
-
v
)
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
}
j
,
k
‖
L
p
(
l
q
)
+
‖
{
2
j
α
2
k
β
∑
j
′
,
k
′
>
0
sup
⁡
(
u
,
v
)
∈
R
n
×
R
m
(
|
ψ
j
,
k
*
θ
j
′
,
k
′
|
*
|
θ
j
′
,
k
′
*
f
|
)
(
⋅
-
u
,
⋅
-
v
)
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
}
j
,
k
‖
L
p
(
l
q
)
:
=
I
1
+
I
2
.We first estimateI
1. By the support properties of ψ
j
,
k and φ and Young’s inequality,(3.8)
I
1
≲
‖
ψ
1,1
*
ϕ
‖
1
⋅
‖
ϕ
*
f
‖
p
≲
‖
ϕ
*
f
‖
p
.Next, we give the estimate forI
2. For any (
u
,
v
)
∈
ℝ
n
×
ℝ
m,(3.9)
(
|
ψ
j
,
k
*
θ
j
′
,
k
′
|
*
|
θ
j
′
,
k
′
*
f
|
)
(
x
-
u
,
y
-
v
)
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
≲
∫
R
n
×
R
m
|
ψ
j
,
k
*
θ
j
′
,
k
′
(
u
′
,
v
′
)
|
|
θ
j
′
,
k
′
*
f
(
x
-
u
-
u
′
,
y
-
v
-
v
′
)
|
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
d
u
′
d
v
′
≤
θ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
∫
R
n
×
R
m
|
ψ
j
,
k
*
θ
j
′
,
k
′
(
u
′
,
v
′
)
|
×
(
1
+
2
j
′
|
u
+
u
′
|
)
b
1
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
′
∧
k
′
|
v
+
v
′
|
)
b
2
(
1
+
2
j
∧
k
|
v
|
)
b
2
d
u
d
v
≲
2
-
L
|
j
-
j
′
|
2
-
L
|
k
-
k
′
|
θ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
,where we have used Lemma 2.2 in the last inequality. Applying Minkowski’s inequality and Hölder’s inequality yields(3.10)
‖
{
2
j
α
2
k
β
∑
j
′
,
k
′
>
0
sup
⁡
(
u
,
v
)
∈
R
n
×
R
m
(
|
ψ
j
,
k
*
θ
j
′
,
k
′
|
*
|
θ
j
′
,
k
′
*
f
|
)
(
x
-
u
,
y
-
v
)
(
1
+
2
j
|
u
|
)
b
1
(
1
+
2
j
∧
k
|
v
|
)
b
2
}
j
,
k
‖
l
q
≲
{
∑
j
′
,
k
′
∈
Z
+
(
∑
j
,
k
∈
Z
+
2
-
|
j
-
j
′
|
q
(
L
-
ɛ
-
|
α
|
)
2
-
|
k
-
k
′
|
q
(
L
-
ɛ
-
|
β
|
)
)
×
2
j
′
α
q
2
k
′
β
q
θ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
q
}
1
/
q
≲
‖
{
2
j
′
α
2
k
′
β
θ
b
⃗
,
j
′
,
k
′
*
(
f
)
(
x
,
y
)
}
j
′
,
k
′
‖
l
q
,where we have chosenɛ as a small positive constant less than L
-
(
|
α
|
∨
|
β
|
). Therefore,(3.11)
I
2
≲
‖
{
2
j
′
α
2
k
′
β
θ
b
⃗
,
j
′
,
k
′
*
(
f
)
}
j
′
,
k
′
‖
L
p
(
l
q
)
.
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 Theorems1.3 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(3.12)
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
≤
2
α
q
2
β
q
‖
ψ
1,1
*
ϕ
‖
1
q
‖
ϕ
*
f
‖
p
q
+
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
(
∑
j
′
=
1
∞
∑
k
′
=
1
∞
‖
ψ
j
,
k
*
θ
j
′
,
k
′
‖
1
‖
θ
j
′
,
k
′
*
f
‖
p
)
q
≲
‖
ϕ
*
f
‖
p
q
+
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
(
∑
j
′
=
1
∞
∑
k
′
=
1
∞
2
-
(
L
-
ɛ
)
q
(
|
j
-
j
′
|
+
|
k
-
k
′
|
)
‖
θ
j
′
k
′
*
f
‖
p
q
)
≲
‖
ϕ
*
f
‖
p
q
+
∑
j
′
=
1
∞
∑
k
′
=
1
∞
2
j
′
α
q
2
k
′
β
q
‖
θ
j
′
,
k
′
*
f
‖
p
q
,as desired. This ends the proof of Theorem1.3 for Besov case. Hence, the proofs of Theorems 1.3 and 1.5 are complete.


## 4. Proof of Theorem1.8

### 4.1. Proof of Theorem1.8 for Besov Space

By the moment conditions ofψ
j
(
1
)’s and ψ
k
(
2
)’s, we may write(4.1)
ψ
j
,
k
*
f
(
x
,
y
)
=
∬
R
n
+
m
×
R
m
ψ
j
(
1
)
(
u
,
v
)
ψ
k
(
2
)
(
w
)
[
(
Δ
-
u
,
-
v
;
-
w
F
)
M
f
(
x
,
y
)
]
d
u
d
v
d
w
.By Minkowski’s inequality and Hölder’s inequality and noting that(4.2)
‖
(
Δ
-
u
,
-
v
;
-
w
F
)
M
f
‖
p
=
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
,we have(4.3)
‖
ψ
j
,
k
*
f
‖
p
q
=
(
∬
R
n
+
m
×
R
m
|
ψ
j
(
1
)
(
u
,
v
)
|
|
ψ
k
(
2
)
(
w
)
|
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
d
u
d
v
d
w
)
q
≲
∬
R
n
+
m
×
R
m
|
ψ
j
(
1
)
(
u
,
v
)
|
|
ψ
k
(
2
)
(
w
)
|
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
q
d
u
d
v
d
w
.Hence,(4.4)
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
≲
∬
R
n
+
m
×
R
m
(
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
)
(
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
)
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
q
d
u
d
v
d
w
≲
‖
f
‖
B
p
,
q
,
(
2
)
α
,
β
q
,where the last inequality follows from(4.5)
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
≲
∑
k
=
1
∞
2
k
β
q
2
-
k
L
(
2
-
k
+
|
w
|
)
m
+
L
≲
1
|
w
|
m
+
β
q
,
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
≲
1
|
(
u
,
v
)
|
n
+
m
+
α
q
.This inequality together with the trivial inequality∥
S
0
f
∥
p
≲
∥
f
∥
p yields ∥
f
∥
ℬ
p
,
q
α
,
β
≲
∥
f
∥
ℬ
p
,
q
,
(
2
)
α
,
β.To prove the converse, by Lemma2.6, it suffices to show that(4.6)
∬
R
n
+
m
×
R
m
(
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
≲
‖
f
‖
B
p
,
q
α
,
β
q
.By the Calderón’s identity (2.1), we write(4.7)
(
Δ
u
,
v
;
w
F
)
M
f
=
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
(
S
0
f
)
+
∑
j
=
1
∞
∑
k
=
1
∞
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
,where the series on the right hand side converges inL
p
(
ℝ
n
+
m
). Thus, by Minkowski’s inequality and Young’s inequality, we conclude that(4.8)
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
≤
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
‖
S
0
f
‖
p
+
∑
j
=
1
∞
∑
k
=
1
∞
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
‖
ψ
j
,
k
*
f
‖
p
.It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of(4.9)
I
I
I
1
≡
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
q
‖
S
0
f
‖
p
q
,
I
I
I
2
≡
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
‖
ψ
j
,
k
*
f
‖
p
)
q
.ForI
I
I
1, we have(4.10)
I
I
I
1
≲
{
∑
j
′
>
0
k
′
>
0
2
-
j
′
q
(
M
-
α
)
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
>
0
k
′
≤
0
2
-
j
′
q
(
M
-
α
)
2
k
′
q
β
+
∑
j
′
≤
0
k
′
>
0
2
j
′
q
α
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
≤
0
k
′
≤
0
2
j
′
q
α
2
k
′
q
α
}
‖
S
0
f
‖
p
q
≲
‖
S
0
f
‖
p
q
,where we have used the estimate(4.11)
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
≲
2
M
⋅
min
⁡
{
0
,
-
j
′
,
-
k
′
,
-
j
′
-
k
′
}
.To estimateI
I
I
2, by the estimate(4.12)
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
≲
2
M
⋅
min
⁡
{
0
,
j
-
j
′
,
k
-
k
′
,
j
-
j
′
+
k
-
k
′
}
,and Hölder’s inequality, we see thatI
I
I
2 is majorized by(4.13)
∑
j
′
,
k
′
∈
Z
(
∑
0
<
j
≤
j
′
0
<
k
≤
k
′
2
q
[
M
-
ɛ
-
(
α
∨
β
)
]
(
j
-
j
′
+
k
-
k
′
)
+
∑
0
<
j
≤
j
′
k
>
(
k
′
∨
0
)
2
q
[
(
M
-
α
-
ɛ
)
(
j
-
j
′
)
+
(
k
′
-
k
)
(
β
-
ɛ
)
]
+
∑
j
>
(
j
′
∨
0
)
0
<
k
≤
k
′
2
q
(
M
-
β
-
ɛ
)
(
k
-
k
′
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
+
∑
j
>
(
j
′
∨
0
)
k
>
(
k
′
∨
0
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
)
×
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
≲
∑
j
>
0
∑
k
>
0
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
,proving (4.6), where ɛ is a positive number such that 0
<
ɛ
<
(
α
∧
β
)
<
(
α
∨
β
)
+
ɛ
<
M. This concludes the proof of Theorem 1.8 for Besov spaces.


### 4.2. Proof of Theorem1.8 for Triebel-Lizorkin Space

Using the moment conditions onψ
j
,
k, we have(4.14)
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
(
x
,
y
)
|
q
≤
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
∬
R
n
+
m
×
R
m
|
ψ
̃
j
(
1
)
(
u
,
v
)
|
|
ψ
̃
k
(
2
)
(
w
)
|
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
q
d
u
d
v
d
w
≤
∬
R
n
+
m
×
R
m
(
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
)
(
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
)
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
q
d
u
d
v
d
w
≲
∬
R
n
+
m
×
R
m
(
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
,whereg
̃
(
x
)
:
=
g
(
-
x
). Therefore,(4.15)
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
|
q
)
1
/
q
‖
p
≲
‖
f
‖
F
p
,
q
,
(
2
)
α
,
β
,which, together with the obvious inequality∥
S
0
f
∥
p
≲
∥
f
∥
p, yields ∥
f
∥
ℱ
p
,
q
α
,
β
≲
∥
f
∥
ℱ
p
,
q
,
(
2
)
α
,
β.To show the converse, write(4.16)
(
Δ
u
,
v
;
w
F
)
M
f
=
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
(
S
0
f
)
+
∑
j
=
1
∞
∑
k
=
1
∞
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
,where the series converges in𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
). It follows that(4.17)
‖
{
∬
R
n
+
m
×
R
m
(
|
(
Δ
u
,
v
;
w
F
)
M
f
|
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
}
1
/
q
‖
p
≤
‖
{
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
|
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
S
0
f
|
q
}
1
/
q
‖
p
+
‖
{
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
|
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
|
)
q
}
1
/
q
‖
p
:
=
I
V
1
+
I
V
2
,where |
(
u
,
v
)
|
~
2
-
j
′ and |
w
|
~
2
-
k
′.ForI
V
2, by Lemma 2.2,(4.18)
|
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
(
u
,
v
)
|
≲
2
M
⋅
min
⁡
{
0
,
j
-
j
′
,
k
-
k
′
,
j
-
j
′
+
k
-
k
′
}
2
j
n
2
(
j
∧
k
)
m
(
1
+
2
j
|
u
|
+
2
j
∧
k
|
v
|
)
L
2
.Consequently,(4.19)
|
(
Δ
u
,
v
;
w
F
)
M
ψ
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
)
.Therefore,(4.20)
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
|
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
(
x
,
y
)
|
)
q
≲
∑
j
,
k
∈
Z
+
(
∑
j
′
>
j
k
′
>
k
2
q
[
M
-
ɛ
-
(
α
∨
β
)
]
(
j
-
j
′
+
k
-
k
′
)
+
∑
j
′
>
j
k
′
<
k
2
q
(
M
-
α
-
ɛ
)
(
j
-
j
′
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
+
∑
j
′
<
j
k
′
>
k
2
q
(
M
-
β
-
ɛ
)
(
k
-
k
′
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
+
∑
j
′
<
j
k
′
<
k
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
)
×
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
q
≲
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
q
,which together with the maximal function characterization ofℱ
p
,
q
α
,
β implies(4.21)
I
V
2
≲
‖
{
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
q
}
1
/
q
‖
p
≲
‖
f
‖
F
p
,
q
α
,
β
.As forI
V
1, similar estimate to (4.19) yields(4.22)
|
(
Δ
u
,
v
;
w
F
)
M
φ
*
S
0
f
(
x
,
y
)
|
≲
2
-
M
⋅
max
⁡
{
0
,
j
′
,
k
′
,
j
′
+
k
′
}
ψ
b
⃗
,
0,0
*
(
f
)
(
x
,
y
)
.Therefore,(4.23)
I
V
1
≲
{
∑
j
′
>
0
k
′
>
0
2
-
j
′
q
(
M
-
α
)
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
>
0
k
′
≤
0
2
-
j
′
q
(
M
-
α
)
2
k
′
q
β
+
∑
j
′
≤
0
k
′
>
0
2
j
′
q
α
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
≤
0
k
′
≤
0
2
j
′
q
α
2
k
′
q
α
}
1
/
q
‖
ψ
b
⃗
,
0,0
*
(
f
)
‖
p
≲
‖
S
0
f
‖
p
.This estimate together with (4.21) and Lemma 2.6 yields ∥
f
∥
ℱ
p
,
q
,
(
2
)
α
,
β
≲
∥
f
∥
ℱ
p
,
q
α
,
β. This ends the proof of Theorem 1.8.


## 4.1. Proof of Theorem1.8 for Besov Space

By the moment conditions ofψ
j
(
1
)’s and ψ
k
(
2
)’s, we may write(4.1)
ψ
j
,
k
*
f
(
x
,
y
)
=
∬
R
n
+
m
×
R
m
ψ
j
(
1
)
(
u
,
v
)
ψ
k
(
2
)
(
w
)
[
(
Δ
-
u
,
-
v
;
-
w
F
)
M
f
(
x
,
y
)
]
d
u
d
v
d
w
.By Minkowski’s inequality and Hölder’s inequality and noting that(4.2)
‖
(
Δ
-
u
,
-
v
;
-
w
F
)
M
f
‖
p
=
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
,we have(4.3)
‖
ψ
j
,
k
*
f
‖
p
q
=
(
∬
R
n
+
m
×
R
m
|
ψ
j
(
1
)
(
u
,
v
)
|
|
ψ
k
(
2
)
(
w
)
|
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
d
u
d
v
d
w
)
q
≲
∬
R
n
+
m
×
R
m
|
ψ
j
(
1
)
(
u
,
v
)
|
|
ψ
k
(
2
)
(
w
)
|
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
q
d
u
d
v
d
w
.Hence,(4.4)
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
≲
∬
R
n
+
m
×
R
m
(
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
)
(
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
)
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
q
d
u
d
v
d
w
≲
‖
f
‖
B
p
,
q
,
(
2
)
α
,
β
q
,where the last inequality follows from(4.5)
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
≲
∑
k
=
1
∞
2
k
β
q
2
-
k
L
(
2
-
k
+
|
w
|
)
m
+
L
≲
1
|
w
|
m
+
β
q
,
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
≲
1
|
(
u
,
v
)
|
n
+
m
+
α
q
.This inequality together with the trivial inequality∥
S
0
f
∥
p
≲
∥
f
∥
p yields ∥
f
∥
ℬ
p
,
q
α
,
β
≲
∥
f
∥
ℬ
p
,
q
,
(
2
)
α
,
β.To prove the converse, by Lemma2.6, it suffices to show that(4.6)
∬
R
n
+
m
×
R
m
(
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
≲
‖
f
‖
B
p
,
q
α
,
β
q
.By the Calderón’s identity (2.1), we write(4.7)
(
Δ
u
,
v
;
w
F
)
M
f
=
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
(
S
0
f
)
+
∑
j
=
1
∞
∑
k
=
1
∞
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
,where the series on the right hand side converges inL
p
(
ℝ
n
+
m
). Thus, by Minkowski’s inequality and Young’s inequality, we conclude that(4.8)
‖
(
Δ
u
,
v
;
w
F
)
M
f
‖
p
≤
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
‖
S
0
f
‖
p
+
∑
j
=
1
∞
∑
k
=
1
∞
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
‖
ψ
j
,
k
*
f
‖
p
.It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of(4.9)
I
I
I
1
≡
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
q
‖
S
0
f
‖
p
q
,
I
I
I
2
≡
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
‖
ψ
j
,
k
*
f
‖
p
)
q
.ForI
I
I
1, we have(4.10)
I
I
I
1
≲
{
∑
j
′
>
0
k
′
>
0
2
-
j
′
q
(
M
-
α
)
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
>
0
k
′
≤
0
2
-
j
′
q
(
M
-
α
)
2
k
′
q
β
+
∑
j
′
≤
0
k
′
>
0
2
j
′
q
α
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
≤
0
k
′
≤
0
2
j
′
q
α
2
k
′
q
α
}
‖
S
0
f
‖
p
q
≲
‖
S
0
f
‖
p
q
,where we have used the estimate(4.11)
‖
(
Δ
u
,
v
;
w
F
)
M
φ
‖
1
≲
2
M
⋅
min
⁡
{
0
,
-
j
′
,
-
k
′
,
-
j
′
-
k
′
}
.To estimateI
I
I
2, by the estimate(4.12)
‖
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
‖
1
≲
2
M
⋅
min
⁡
{
0
,
j
-
j
′
,
k
-
k
′
,
j
-
j
′
+
k
-
k
′
}
,and Hölder’s inequality, we see thatI
I
I
2 is majorized by(4.13)
∑
j
′
,
k
′
∈
Z
(
∑
0
<
j
≤
j
′
0
<
k
≤
k
′
2
q
[
M
-
ɛ
-
(
α
∨
β
)
]
(
j
-
j
′
+
k
-
k
′
)
+
∑
0
<
j
≤
j
′
k
>
(
k
′
∨
0
)
2
q
[
(
M
-
α
-
ɛ
)
(
j
-
j
′
)
+
(
k
′
-
k
)
(
β
-
ɛ
)
]
+
∑
j
>
(
j
′
∨
0
)
0
<
k
≤
k
′
2
q
(
M
-
β
-
ɛ
)
(
k
-
k
′
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
+
∑
j
>
(
j
′
∨
0
)
k
>
(
k
′
∨
0
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
)
×
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
≲
∑
j
>
0
∑
k
>
0
2
j
α
q
2
k
β
q
‖
ψ
j
,
k
*
f
‖
p
q
,proving (4.6), where ɛ is a positive number such that 0
<
ɛ
<
(
α
∧
β
)
<
(
α
∨
β
)
+
ɛ
<
M. This concludes the proof of Theorem 1.8 for Besov spaces.


## 4.2. Proof of Theorem1.8 for Triebel-Lizorkin Space

Using the moment conditions onψ
j
,
k, we have(4.14)
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
(
x
,
y
)
|
q
≤
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
∬
R
n
+
m
×
R
m
|
ψ
̃
j
(
1
)
(
u
,
v
)
|
|
ψ
̃
k
(
2
)
(
w
)
|
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
q
d
u
d
v
d
w
≤
∬
R
n
+
m
×
R
m
(
∑
j
=
1
∞
2
j
α
q
|
ψ
j
(
1
)
(
u
,
v
)
|
)
(
∑
k
=
1
∞
2
k
β
q
|
ψ
k
(
2
)
(
w
)
|
)
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
q
d
u
d
v
d
w
≲
∬
R
n
+
m
×
R
m
(
|
(
Δ
u
,
v
;
w
F
)
M
f
(
x
,
y
)
|
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
,whereg
̃
(
x
)
:
=
g
(
-
x
). Therefore,(4.15)
‖
(
∑
j
=
1
∞
∑
k
=
1
∞
2
j
α
q
2
k
β
q
|
ψ
j
,
k
*
f
|
q
)
1
/
q
‖
p
≲
‖
f
‖
F
p
,
q
,
(
2
)
α
,
β
,which, together with the obvious inequality∥
S
0
f
∥
p
≲
∥
f
∥
p, yields ∥
f
∥
ℱ
p
,
q
α
,
β
≲
∥
f
∥
ℱ
p
,
q
,
(
2
)
α
,
β.To show the converse, write(4.16)
(
Δ
u
,
v
;
w
F
)
M
f
=
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
(
S
0
f
)
+
∑
j
=
1
∞
∑
k
=
1
∞
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
,where the series converges in𝒮
𝔉
′
(
ℝ
n
×
ℝ
m
). It follows that(4.17)
‖
{
∬
R
n
+
m
×
R
m
(
|
(
Δ
u
,
v
;
w
F
)
M
f
|
|
(
u
,
v
)
|
α
|
w
|
β
)
q
d
u
d
v
d
w
|
(
u
,
v
)
|
n
+
m
|
w
|
m
}
1
/
q
‖
p
≤
‖
{
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
|
(
(
Δ
u
,
v
;
w
F
)
M
φ
)
*
S
0
f
|
q
}
1
/
q
‖
p
+
‖
{
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
|
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
|
)
q
}
1
/
q
‖
p
:
=
I
V
1
+
I
V
2
,where |
(
u
,
v
)
|
~
2
-
j
′ and |
w
|
~
2
-
k
′.ForI
V
2, by Lemma 2.2,(4.18)
|
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
(
u
,
v
)
|
≲
2
M
⋅
min
⁡
{
0
,
j
-
j
′
,
k
-
k
′
,
j
-
j
′
+
k
-
k
′
}
2
j
n
2
(
j
∧
k
)
m
(
1
+
2
j
|
u
|
+
2
j
∧
k
|
v
|
)
L
2
.Consequently,(4.19)
|
(
Δ
u
,
v
;
w
F
)
M
ψ
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
)
.Therefore,(4.20)
∑
j
′
∈
Z
∑
k
′
∈
Z
2
j
′
α
q
2
k
′
β
q
(
∑
j
=
1
∞
∑
k
=
1
∞
|
(
(
Δ
u
,
v
;
w
F
)
M
ψ
j
,
k
)
*
ψ
j
,
k
*
f
(
x
,
y
)
|
)
q
≲
∑
j
,
k
∈
Z
+
(
∑
j
′
>
j
k
′
>
k
2
q
[
M
-
ɛ
-
(
α
∨
β
)
]
(
j
-
j
′
+
k
-
k
′
)
+
∑
j
′
>
j
k
′
<
k
2
q
(
M
-
α
-
ɛ
)
(
j
-
j
′
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
+
∑
j
′
<
j
k
′
>
k
2
q
(
M
-
β
-
ɛ
)
(
k
-
k
′
)
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
+
∑
j
′
<
j
k
′
<
k
2
q
(
j
′
-
j
)
(
α
-
ɛ
)
2
q
(
k
′
-
k
)
(
β
-
ɛ
)
)
×
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
q
≲
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
(
x
,
y
)
q
,which together with the maximal function characterization ofℱ
p
,
q
α
,
β implies(4.21)
I
V
2
≲
‖
{
∑
j
,
k
∈
Z
+
2
j
α
q
2
k
β
q
ψ
b
⃗
,
j
,
k
*
(
f
)
q
}
1
/
q
‖
p
≲
‖
f
‖
F
p
,
q
α
,
β
.As forI
V
1, similar estimate to (4.19) yields(4.22)
|
(
Δ
u
,
v
;
w
F
)
M
φ
*
S
0
f
(
x
,
y
)
|
≲
2
-
M
⋅
max
⁡
{
0
,
j
′
,
k
′
,
j
′
+
k
′
}
ψ
b
⃗
,
0,0
*
(
f
)
(
x
,
y
)
.Therefore,(4.23)
I
V
1
≲
{
∑
j
′
>
0
k
′
>
0
2
-
j
′
q
(
M
-
α
)
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
>
0
k
′
≤
0
2
-
j
′
q
(
M
-
α
)
2
k
′
q
β
+
∑
j
′
≤
0
k
′
>
0
2
j
′
q
α
2
-
k
′
q
(
M
-
β
)
+
∑
j
′
≤
0
k
′
≤
0
2
j
′
q
α
2
k
′
q
α
}
1
/
q
‖
ψ
b
⃗
,
0,0
*
(
f
)
‖
p
≲
‖
S
0
f
‖
p
.This estimate together with (4.21) and Lemma 2.6 yields ∥
f
∥
ℱ
p
,
q
,
(
2
)
α
,
β
≲
∥
f
∥
ℱ
p
,
q
α
,
β. This ends the proof of Theorem 1.8.



---

*Source: 275791-2012-05-13.xml*