# Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Authors: Pu Zhang; Daiqing Zhang
Journal: The Scientific World Journal
 (2014)

---

## Abstract

LetT be a singular integral operator with its kernel satisfying |
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
)
ϕ
k
(
y
)
|
≤
C
|
y
|
γ
/
|
x
-
y
|
n
+
γ, |
x
|
>
2
|
y
|
>
0, where B
k and ϕ
k
(
k
=
1
,
…
,
ℓ
) are appropriate functions and γ and C are positive constants. For b
→
=
(
b
1
,
…
,
b
m
) with b
j
∈
BMO
(
ℝ
n
), the multilinear commutator T
b
→ generated by T and b
→ is formally defined by T
b
→
f
(
x
)
=
∫
ℝ
n
∏
j
=
1
m
‍
(
b
j
(
x
)
-
b
j
(
y
)
)
K
(
x
,
y
)
f
(
y
)
d
y. In this paper, the weighted L
p-boundedness and the weighted weak type L
log
⁡
L estimate for the multilinear commutator T
b
→ are established.


---

## Body

## 1. Introduction and Results

In the classical Calderón-Zygmund theory, the Hörmander's condition(1)
∫
|
x
|
>
2
|
y
|
‍
|
K
(
x
-
y
)
-
K
(
x
)
|
d
x
≤
C
,
introduced by Hörmander [1], plays a fundamental role in the theory of Calderón-Zygmund operators. On the other hand, singular integral operators whose kernels do not satisfy the Hörmander's condition have been extensively studied.In 1997, in order to study theL
p-boundedness of certain singular integral operators, Grubb and Moore [2] introduced the following variant of the classical Hörmander's condition,
(2)
∫
|
x
|
>
2
|
y
|
‍
|
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
)
ϕ
k
(
y
)
|
d
x
≤
C
,
where B
k and ϕ
k's are appropriate functions (see Theorem 3 below). As an example we note that the kernel K
(
x
)
=
sin
x
/
x verifies (2), but it is not a Calderón-Zygmund kernel since its derivative does not decay quickly enough at infinity (see [2] or [3]).Obviously, if we takeℓ
=
1, B
1
(
x
)
=
K
(
x
) and ϕ
1
(
y
)
≡
1, then condition (2) is exactly the classical Hörmander's condition (1).Definition 1.
We say that a nonnegative locally integrable functiong defined on ℝ
n satisfies the reverse Hölder R
H
∞ condition, in short, g
∈
R
H
∞
(
ℝ
n
), if there is a constant C
>
0 such that for every cube Q
⊂
ℝ
n centered at the origin we have
(3)
0
<
sup
⁡
x
∈
Q
g
(
x
)
≤
C
1
|
Q
|
∫
Q
‍
g
(
x
)
d
x
.
The smallest constant C is said to be the R
H
∞ constant of g.Remark 2.
It is easy to see that ifg
(
x
)
∈
R
H
∞
(
ℝ
n
), then also g
(
-
x
)
∈
R
H
∞
(
ℝ
n
) (see [3] Remark 2.4).In [2], Grubb and Moore established the L
p-boundedness and the weak type (
1,1
) estimates for the singular integral operators with kernels satisfying (2).It is well known that the classical Hörmander's condition (1) is too weak to get weighted inequalities for the classical Calderón-Zygmund operators by any known method. The usual hypothesis on the kernel K to obtain them is the Lipschitz condition
(4)
|
K
(
x
-
y
)
-
K
(
x
)
|
≤
C
|
y
|
γ
|
x
-
y
|
n
+
γ
,
|
x
|
>
c
|
y
|
.
Conditions, the so-called L
r-Hörmander's condition, weaker than (4), but stronger than (1), have been also considered in [4, 5] (also see [6, 7]).In 2003, Trujillo-González [3] establishes the weighted norm inequalities for T when K satisfies a variant of the Lipschitz condition (see (6) below).As usual, we denote byA
p
(
1
≤
p
≤
∞
) the Muckenhoupt weights classes (see [8], or [9] and [10]). For a weight ω, 1
≤
p
<
∞ and a measurable set E, we write
(5)
∥
f
∥
L
p
(
ω
)
=
(
∫
ℝ
n
‍
|
f
(
x
)
|
p
ω
(
x
)
d
x
)
1
/
p
,
ω
(
E
)
=
∫
E
‍
ω
(
x
)
d
x
.Theorem 3 (see [3]).
LetK
∈
L
2
(
ℝ
n
). Suppose that there is a constant C
0
>
0, such that(
K
1
)
∥
K
^
∥
∞
≤
C
0;
(
K
2
)
|
K
(
x
)
|
≤
C
0
|
x
|
-
n;
(
K
3
)
there exist functionsB
1
,
…
,
B
ℓ
∈
L
loc
⁡
1
(
ℝ
n
∖
{
0
}
) and {
ϕ
1
,
…
,
ϕ
ℓ
}
⊂
L
∞
(
ℝ
n
) such that |
det
⁡
[
ϕ
k
(
y
i
)
]
|
2
∈
R
H
∞
(
ℝ
n
ℓ
), where y
i
∈
ℝ
n and i
,
k
=
1
,
…
,
ℓ;
(
K
4
)
for a fixedγ
>
0 and for any |
x
|
>
2
|
y
|
>
0,
(6)
|
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
)
ϕ
k
(
y
)
|
≤
C
0
|
y
|
γ
|
x
-
y
|
n
+
γ
.
Forf
∈
C
0
∞
(
ℝ
n
), we defined the convolution operator associated to the kernel K by
(7)
T
f
(
x
)
=
∫
ℝ
n
‍
K
(
x
-
y
)
f
(
y
)
d
y
.
(1)
Let1
<
p
<
∞ and ω
∈
A
p. Then there exists a constant C
>
0 such that
(8)
∫
ℝ
n
‍
|
T
f
(
x
)
|
p
ω
(
x
)
d
x
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
p
ω
(
x
)
d
x
.
(2)
Letω
∈
A
1. Then there exists a constant C
>
0 such that for all λ
>
0
(9)
ω
(
{
x
∈
ℝ
n
:
|
T
f
(
x
)
|
>
λ
}
)
≤
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.It is easy to see that any kernel satisfies condition (6) and also verifies (2). Obviously, if we take ℓ
=
1, B
1
(
x
)
=
K
(
x
), and ϕ
1
(
y
)
≡
1, then condition (6) is exactly the classical Lipschitz condition (4). We remark that the function K
(
x
)
=
sin
x
/
x satisfies conditions (
K
1
)
–
(
K
4
), but does not satisfy the Hörmander's condition (1) (see [11] page 5).Under the assumption of Theorem3, several authors have studied two-weight inequalities for the convolution operator T, for example [11–13]. Recently, the authors [14] introduce a variant of the classical L
r-Hörmander's condition in the scope of (2) and establish the weighted norm inequalities for singular integral operator with its kernel satisfying such a variant of the classical L
r-Hörmander's condition.On the other hand, the commutators of singular integral operators have been widely studied by many authors; see, for example, [15–22] and the references therein. Given a locally integrable function b and a linear operator T with kernel K, the linear commutator [
b
,
T
] is formally defined by
(10)
[
b
,
T
]
f
=
b
T
(
f
)
-
T
(
b
f
)
.Forb
→
=
(
b
1
,
…
,
b
m
) with b
j
∈
B
M
O
(
ℝ
n
)
(
j
=
1
,
…
,
m
). The generalized commutator, the so-called the multilinear commutator, T
b
→ is formally defined by
(11)
T
b
→
f
(
x
)
=
∫
ℝ
n
‍
[
∏
i
=
1
m
‍
(
b
j
(
x
)
-
b
j
(
y
)
)
]
K
(
x
,
y
)
f
(
y
)
d
y
.In 2002, Pérez and Trujillo-González [22] studied the sharp weighted estimates for the multilinear commutators of the classical Calderón-Zygmund operators. In 2006, Zhang [23] studied the weighted estimates for maximal multilinear commutators.In 1993, Alvarez et al. [15] established a generalized boundedness criterion for the commutators of linear operators. Now, we restate Theorem 2.13 in [15] in the following strong form.Theorem 4 (see [15]).
Let𝒦 be a linear operator and 1
<
p
<
∞. Suppose that for all ω
∈
A
p
(
ℝ
n
), the linear operator 𝒦 satisfies the following weighted estimate
(12)
∥
𝒦
f
∥
L
p
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
,
where the constant C depends only on n, p, and the A
p constant of ω. Then for b
∈
B
M
O
(
ℝ
n
) and any weight function ν
∈
A
p, the commutator [
b
,
𝒦
] is bounded from L
p
(
ν
) to L
p
(
ν
) with bound depending on n, p, and the A
p constant of ω.The goal of this paper is to study the weighted norm inequalities for multilinear commutator of the convolution operatorT defined by (7) with its kernel satisfying (
K
1
)
–
(
K
4
).By Theorem3 and applying Theorem 4
m-times, we can easily get the following weighted L
p inequalities for the multilinear commutator T
b
→.Theorem 5.
LetT be the singular integral operator defined by (7) with its kernel satisfying (
K
1
)
–
(
K
4
). If 1
<
p
<
∞, ω
∈
A
p, and b
j
∈
B
M
O
(
ℝ
n
)
(
j
=
1
,
…
,
m
), then there exists a positive constant C such that
(13)
∫
ℝ
n
‍
|
T
b
→
f
(
x
)
|
p
ω
(
x
)
d
x
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
p
ω
(
x
)
d
x
.It is well-known that, in general, the linear commutator of Calderón-Zygmund operator fails to be of weak type(
1,1
) and does not map H
1
(
ℝ
n
) into L
1
(
ℝ
n
) when b
∈
B
M
O
(
ℝ
n
); see [20] for more details. Instead, an endpoint theory was provided for this operator, such as the weak type L
log
⁡
L estimate and the weak type (
H
1
,
L
1
) estimate (see [20, 24]).The main result of this paper is the following weak typeL
log
⁡
L estimate for multilinear commutator of the singular integral operator defined in Theorem 3.Theorem 6.
LetT be the singular integral operator defined by (7) with its kernel satisfying (
K
1
)
–
(
K
4
). If ω
∈
A
1 and b
j
∈
B
M
O
(
ℝ
n
)
(
j
=
1
,
…
,
m
), then, for all λ
>
0,
(14)
ω
(
{
x
∈
ℝ
n
:
|
T
b
→
f
(
x
)
|
>
λ
}
)
≤
C
∫
ℝ
n
‍
|
f
(
y
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
m
ω
(
y
)
d
y
,
where C is a positive constant independent of λ and f.Throughout this paper,γ denotes the positive number appeared in (6). As usual, the letter C stands for a positive constant which is independent of the main parameters and not necessary the same at each occurrence. A cube Q in ℝ
n always means a cube whose sides parallel to the coordinate axes. For a cube Q and a number t
>
0, we denote by t
Q the cube with the same center and t-times the side length as Q. The symbol A
≈
B means there exist positive constants C
1 and C
2 such that C
1
A
≤
B
≤
C
2
A.This paper is arranged as follows. In Section2, we formulate some preliminaries and lemmas we need. In Section 3 we will prove Theorem 6 for the case m
=
1, and in the last section we prove Theorem 6 for the general case m
>
1.


## 2. Preliminaries and Lemmas

In this section, we give some notations and results needed for the proof of the main result.


### 2.1. Muckenhoupt Weight Classes

A nonnegative locally integrable function defined onℝ
n is called a weight. We say a weight ∈
A
p
(
1
<
p
<
∞
), if there exists a constant C
>
0 such that for all cubes Q
⊂
ℝ
n
(15)
(
1
|
Q
|
∫
Q
‍
ω
(
x
)
d
x
)
(
1
|
Q
|
∫
Q
‍
ω
(
x
)
-
1
/
(
p
-
1
)
d
x
)
p
-
1
≤
C
.We say a weightω
∈
A
1, if there exists a constant C
>
0 such that for all cubes Q
⊂
ℝ
n
(16)
1
|
Q
|
∫
Q
‍
ω
(
y
)
d
y
≤
C
ess
inf
⁡
y
∈
Q
⁡
ω
(
y
)
.TheA
∞ weights class is defined by A
∞
=
⋃
1
<
p
<
∞
A
p. There is also another characterization of the A
∞ class, that is, we say a weight ω
∈
A
∞, if there exist positive constants C and δ such that, for any cube Q and any measurable set E
⊂
Q, there exist
(17)
ω
(
E
)
ω
(
Q
)
≤
C
(
|
E
|
|
Q
|
)
δ
.


### 2.2. Projection of Function

Now, let us recall the definition of the projection of a function (see [2] or [3]). By the projection of an L
1-function f onto a finite-dimensional subspace Y we refer to such an element, if it exists P
(
f
) of Y verifying
(18)
∫
f
(
x
)
h
¯
(
x
)
d
x
=
∫
P
(
f
)
(
x
)
h
¯
(
x
)
d
x
,
for
every
h
∈
Y
.Lemma 7 (see [2]).
Suppose{
ϕ
1
,
…
,
ϕ
ℓ
} is a finite family of bounded functions on ℝ
n such that |
det
⁡
[
ϕ
k
(
y
i
)
]
|
2
∈
R
H
∞
(
ℝ
n
ℓ
). Then, for any cube Q centered at the origin and any f
∈
L
1
(
Q
), there exists the projection P
Q
f of f onto span
⁡
{
ϕ
1
,
…
,
ϕ
ℓ
}
⊂
L
1
(
Q
) and satisfies
(19)
sup
⁡
y
∈
Q
|
P
Q
f
(
y
)
|
≤
C
1
|
Q
|
∫
Q
‍
|
f
(
y
)
|
d
y
,
where the constant C depends only on n, ℓ, and the R
H
∞ constant of |
det
⁡
[
ϕ
k
(
y
i
)
]
|
2.


### 2.3. Notations Related to Orlicz Spaces

A functionΦ
:
[
0
,
∞
)
→
[
0
,
∞
) is said to be a Young function, if Φ is continuous, convex, and increasing with Φ
(
0
)
=
0 and lim
⁡
t
→
∞
Φ
(
t
)
=
∞. We use Φ
~ to denote the complementary Young function associated to Φ; that is,
(20)
Φ
~
(
s
)
=
sup
⁡
0
≤
t
<
∞
{
s
t
-
Φ
(
t
)
}
,
0
≤
s
<
∞
.TheΦ-average of a locally integrable function f over a cube Q
⊂
ℝ
n is defined by
(21)
∥
f
∥
Φ
,
Q
=
inf
⁡
{
λ
>
0
:
1
|
Q
|
∫
Q
‍
Φ
(
|
f
(
y
)
|
λ
)
d
y
≤
1
}
,
which satisfies the following inequalities (see [25], p. 69, or formula (
7
) in [21]):
(22)
∥
f
∥
Φ
,
Q
≤
inf
⁡
η
>
0
{
η
+
η
|
Q
|
∫
Q
‍
Φ
(
|
f
(
y
)
|
η
)
d
y
}
≤
2
∥
f
∥
Φ
,
Q
.The Young function that we are going to use isΦ
α
(
t
)
=
t
(
1
+
log
⁡
+
t
)
α
(
α
>
0
) with its complementary Young function Φ
~
α
(
t
)
≈
exp
⁡
(
t
1
/
α
). Denote
(23)
∥
f
∥
L
(
log
⁡
L
)
α
,
Q
=
∥
f
∥
Φ
α
,
Q
,
∥
f
∥
exp
⁡
L
1
/
α
,
Q
=
∥
f
∥
Φ
~
α
,
Q
.Whenα
=
1, we simply write Φ
(
t
)
=
t
(
1
+
log
⁡
+
t
) and Φ
~
(
t
)
≈
e
t, and ∥
f
∥
L
(
log
⁡
L
)
,
Q
=
∥
f
∥
Φ
,
Q and ∥
f
∥
exp
⁡
L
,
Q
=
∥
f
∥
Φ
~
,
Q.The following generalized Hölder’s inequality holds (see(
2.5
) in [22]):
(24)
1
|
Q
|
∫
Q
‍
|
f
1
(
y
)
f
2
(
y
)
⋯
f
m
(
y
)
g
(
y
)
|
d
y
≤
C
∥
g
∥
L
(
log
⁡
L
)
m
,
Q
∏
j
=
1
m
‍
∥
f
j
∥
exp
⁡
L
,
Q
.We also need the following notations (see [26] pages 1712-1713). For ω
∈
A
∞ and a cube Q
⊂
ℝ
n, denote
(25)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
=
inf
⁡
{
λ
>
0
:
1
ω
(
Q
)
∫
Q
‍
Φ
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
≤
1
}
,
∥
f
∥
exp
⁡
L
1
/
m
,
Q
,
ω
=
inf
⁡
{
λ
>
0
:
1
ω
(
Q
)
∫
Q
‍
Φ
~
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
≤
1
}
.Similarly to (22), we have
(26)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
≈
inf
⁡
η
>
0
{
η
+
η
ω
(
Q
)
∫
Q
‍
Φ
m
(
|
f
(
y
)
|
η
)
ω
(
y
)
d
y
}
.There also holds the following generalized Hölder's inequality:(27)
1
ω
(
Q
)
∫
Q
‍
|
f
1
(
y
)
⋯
f
m
(
y
)
g
(
y
)
|
ω
(
y
)
d
y
≤
C
∥
g
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
∏
j
=
1
m
‍
∥
f
j
∥
exp
⁡
L
,
Q
,
ω
.


### 2.4. Lemmas

The following generalized Young's inequality is from [22] Lemma 8. We note that when k
=
2, it is proved by O'Neil in [27].Lemma 8 (the generalized Young's inequality).
φ
0
,
φ
1
,
…
φ
k are real-valued, nonnegative, nondecreasing, left continuous functions defined on [
0
,
∞
). For 0
≤
t
<
∞, define φ
j
-
1
(
t
)
=
inf
⁡
{
s
:
φ
j
(
s
)
>
t
}
(
j
=
0,1
,
…
,
k
). If for all 0
≤
t
<
∞
(28)
φ
1
-
1
(
t
)
⋯
φ
k
-
1
(
t
)
≤
φ
0
-
1
(
t
)
.
Then, for all 0
≤
t
1
,
t
2
,
…
,
t
k
<
∞, there exist
(29)
φ
0
(
t
1
t
2
⋯
t
k
)
≤
φ
1
(
t
1
)
+
φ
1
(
t
2
)
+
⋯
+
φ
k
(
t
k
)
.ForΦ
k
(
t
)
=
t
(
1
+
log
⁡
+
t
)
k
(
k
=
1
,
…
,
m
) and Ψ
(
t
)
=
e
t
-
1, we have Φ
k
-
1
(
t
)
≈
t
/
(
log
⁡
t
)
k and Ψ
-
1
(
t
)
≈
log
⁡
t (see [21] page 35). Then for any integer j with 1
≤
j
≤
m
-
1, we have
(30)
Φ
m
-
1
(
t
)
Ψ
-
1
(
t
)
⋯
Ψ
-
1
(
t
)
︸
m
-
j
≤
C
Φ
j
-
1
(
t
)
:
=
𝒜
-
1
(
t
)
.Noting that𝒜
(
t
)
=
Φ
j
(
C
-
1
t
) since 𝒜
-
1
(
t
)
=
C
Φ
j
-
1
(
t
), then it follows from Lemma 8 that, for all 0
≤
s, t
1
,
t
2
,
…
,
t
m
-
j
<
∞, we have
(31)
Φ
j
(
C
-
1
s
·
t
1
⋯
t
m
-
j
)
=
𝒜
(
s
·
t
1
⋯
t
m
-
j
)
≤
Φ
m
(
s
)
+
Ψ
(
t
1
)
+
⋯
+
Ψ
(
t
m
-
j
)
.For a locally integrable functionf and a cube Q, denote
(32)
f
Q
=
(
f
)
Q
=
1
|
Q
|
∫
Q
‍
f
(
y
)
d
y
.Lemma 9 (see [26]).
Letω
∈
A
∞ and b
∈
B
M
O
(
ℝ
n
). Then, for any cube Q
⊂
ℝ
n,
(33)
1
ω
(
Q
)
∫
Q
‍
exp
⁡
(
|
b
(
x
)
-
b
Q
|
C
0
∥
b
∥
*
)
ω
(
x
)
d
x
≤
C
,
∥
b
-
b
Q
∥
exp
⁡
L
,
Q
,
ω
≤
C
∥
b
∥
*
,
where C
0 and C are positive constants independent of b and Q, and ∥
b
∥
* is the B
M
O-norm of b.Lemma 10 (see [28]).
Let1
≤
p
<
∞, ω
p
∈
A
1, b
j
∈
B
M
O
(
ℝ
n
)
(
j
=
1
,
…
,
m
), and Q be a cube. Then for any positive integer m and k
=
0,1
,
…,
(34)
(
1
|
2
k
Q
|
∫
2
k
Q
‍
ω
p
(
x
)
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
|
p
d
x
)
1
/
p
≤
C
∥
b
→
∥
*
(
k
+
1
)
m
ess
inf
⁡
y
∈
Q
⁡
ω
(
y
)
.


## 2.1. Muckenhoupt Weight Classes

A nonnegative locally integrable function defined onℝ
n is called a weight. We say a weight ∈
A
p
(
1
<
p
<
∞
), if there exists a constant C
>
0 such that for all cubes Q
⊂
ℝ
n
(15)
(
1
|
Q
|
∫
Q
‍
ω
(
x
)
d
x
)
(
1
|
Q
|
∫
Q
‍
ω
(
x
)
-
1
/
(
p
-
1
)
d
x
)
p
-
1
≤
C
.We say a weightω
∈
A
1, if there exists a constant C
>
0 such that for all cubes Q
⊂
ℝ
n
(16)
1
|
Q
|
∫
Q
‍
ω
(
y
)
d
y
≤
C
ess
inf
⁡
y
∈
Q
⁡
ω
(
y
)
.TheA
∞ weights class is defined by A
∞
=
⋃
1
<
p
<
∞
A
p. There is also another characterization of the A
∞ class, that is, we say a weight ω
∈
A
∞, if there exist positive constants C and δ such that, for any cube Q and any measurable set E
⊂
Q, there exist
(17)
ω
(
E
)
ω
(
Q
)
≤
C
(
|
E
|
|
Q
|
)
δ
.


## 2.2. Projection of Function

Now, let us recall the definition of the projection of a function (see [2] or [3]). By the projection of an L
1-function f onto a finite-dimensional subspace Y we refer to such an element, if it exists P
(
f
) of Y verifying
(18)
∫
f
(
x
)
h
¯
(
x
)
d
x
=
∫
P
(
f
)
(
x
)
h
¯
(
x
)
d
x
,
for
every
h
∈
Y
.Lemma 7 (see [2]).
Suppose{
ϕ
1
,
…
,
ϕ
ℓ
} is a finite family of bounded functions on ℝ
n such that |
det
⁡
[
ϕ
k
(
y
i
)
]
|
2
∈
R
H
∞
(
ℝ
n
ℓ
). Then, for any cube Q centered at the origin and any f
∈
L
1
(
Q
), there exists the projection P
Q
f of f onto span
⁡
{
ϕ
1
,
…
,
ϕ
ℓ
}
⊂
L
1
(
Q
) and satisfies
(19)
sup
⁡
y
∈
Q
|
P
Q
f
(
y
)
|
≤
C
1
|
Q
|
∫
Q
‍
|
f
(
y
)
|
d
y
,
where the constant C depends only on n, ℓ, and the R
H
∞ constant of |
det
⁡
[
ϕ
k
(
y
i
)
]
|
2.


## 2.3. Notations Related to Orlicz Spaces

A functionΦ
:
[
0
,
∞
)
→
[
0
,
∞
) is said to be a Young function, if Φ is continuous, convex, and increasing with Φ
(
0
)
=
0 and lim
⁡
t
→
∞
Φ
(
t
)
=
∞. We use Φ
~ to denote the complementary Young function associated to Φ; that is,
(20)
Φ
~
(
s
)
=
sup
⁡
0
≤
t
<
∞
{
s
t
-
Φ
(
t
)
}
,
0
≤
s
<
∞
.TheΦ-average of a locally integrable function f over a cube Q
⊂
ℝ
n is defined by
(21)
∥
f
∥
Φ
,
Q
=
inf
⁡
{
λ
>
0
:
1
|
Q
|
∫
Q
‍
Φ
(
|
f
(
y
)
|
λ
)
d
y
≤
1
}
,
which satisfies the following inequalities (see [25], p. 69, or formula (
7
) in [21]):
(22)
∥
f
∥
Φ
,
Q
≤
inf
⁡
η
>
0
{
η
+
η
|
Q
|
∫
Q
‍
Φ
(
|
f
(
y
)
|
η
)
d
y
}
≤
2
∥
f
∥
Φ
,
Q
.The Young function that we are going to use isΦ
α
(
t
)
=
t
(
1
+
log
⁡
+
t
)
α
(
α
>
0
) with its complementary Young function Φ
~
α
(
t
)
≈
exp
⁡
(
t
1
/
α
). Denote
(23)
∥
f
∥
L
(
log
⁡
L
)
α
,
Q
=
∥
f
∥
Φ
α
,
Q
,
∥
f
∥
exp
⁡
L
1
/
α
,
Q
=
∥
f
∥
Φ
~
α
,
Q
.Whenα
=
1, we simply write Φ
(
t
)
=
t
(
1
+
log
⁡
+
t
) and Φ
~
(
t
)
≈
e
t, and ∥
f
∥
L
(
log
⁡
L
)
,
Q
=
∥
f
∥
Φ
,
Q and ∥
f
∥
exp
⁡
L
,
Q
=
∥
f
∥
Φ
~
,
Q.The following generalized Hölder’s inequality holds (see(
2.5
) in [22]):
(24)
1
|
Q
|
∫
Q
‍
|
f
1
(
y
)
f
2
(
y
)
⋯
f
m
(
y
)
g
(
y
)
|
d
y
≤
C
∥
g
∥
L
(
log
⁡
L
)
m
,
Q
∏
j
=
1
m
‍
∥
f
j
∥
exp
⁡
L
,
Q
.We also need the following notations (see [26] pages 1712-1713). For ω
∈
A
∞ and a cube Q
⊂
ℝ
n, denote
(25)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
=
inf
⁡
{
λ
>
0
:
1
ω
(
Q
)
∫
Q
‍
Φ
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
≤
1
}
,
∥
f
∥
exp
⁡
L
1
/
m
,
Q
,
ω
=
inf
⁡
{
λ
>
0
:
1
ω
(
Q
)
∫
Q
‍
Φ
~
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
≤
1
}
.Similarly to (22), we have
(26)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
≈
inf
⁡
η
>
0
{
η
+
η
ω
(
Q
)
∫
Q
‍
Φ
m
(
|
f
(
y
)
|
η
)
ω
(
y
)
d
y
}
.There also holds the following generalized Hölder's inequality:(27)
1
ω
(
Q
)
∫
Q
‍
|
f
1
(
y
)
⋯
f
m
(
y
)
g
(
y
)
|
ω
(
y
)
d
y
≤
C
∥
g
∥
L
(
log
⁡
L
)
m
,
Q
,
ω
∏
j
=
1
m
‍
∥
f
j
∥
exp
⁡
L
,
Q
,
ω
.


## 2.4. Lemmas

The following generalized Young's inequality is from [22] Lemma 8. We note that when k
=
2, it is proved by O'Neil in [27].Lemma 8 (the generalized Young's inequality).
φ
0
,
φ
1
,
…
φ
k are real-valued, nonnegative, nondecreasing, left continuous functions defined on [
0
,
∞
). For 0
≤
t
<
∞, define φ
j
-
1
(
t
)
=
inf
⁡
{
s
:
φ
j
(
s
)
>
t
}
(
j
=
0,1
,
…
,
k
). If for all 0
≤
t
<
∞
(28)
φ
1
-
1
(
t
)
⋯
φ
k
-
1
(
t
)
≤
φ
0
-
1
(
t
)
.
Then, for all 0
≤
t
1
,
t
2
,
…
,
t
k
<
∞, there exist
(29)
φ
0
(
t
1
t
2
⋯
t
k
)
≤
φ
1
(
t
1
)
+
φ
1
(
t
2
)
+
⋯
+
φ
k
(
t
k
)
.ForΦ
k
(
t
)
=
t
(
1
+
log
⁡
+
t
)
k
(
k
=
1
,
…
,
m
) and Ψ
(
t
)
=
e
t
-
1, we have Φ
k
-
1
(
t
)
≈
t
/
(
log
⁡
t
)
k and Ψ
-
1
(
t
)
≈
log
⁡
t (see [21] page 35). Then for any integer j with 1
≤
j
≤
m
-
1, we have
(30)
Φ
m
-
1
(
t
)
Ψ
-
1
(
t
)
⋯
Ψ
-
1
(
t
)
︸
m
-
j
≤
C
Φ
j
-
1
(
t
)
:
=
𝒜
-
1
(
t
)
.Noting that𝒜
(
t
)
=
Φ
j
(
C
-
1
t
) since 𝒜
-
1
(
t
)
=
C
Φ
j
-
1
(
t
), then it follows from Lemma 8 that, for all 0
≤
s, t
1
,
t
2
,
…
,
t
m
-
j
<
∞, we have
(31)
Φ
j
(
C
-
1
s
·
t
1
⋯
t
m
-
j
)
=
𝒜
(
s
·
t
1
⋯
t
m
-
j
)
≤
Φ
m
(
s
)
+
Ψ
(
t
1
)
+
⋯
+
Ψ
(
t
m
-
j
)
.For a locally integrable functionf and a cube Q, denote
(32)
f
Q
=
(
f
)
Q
=
1
|
Q
|
∫
Q
‍
f
(
y
)
d
y
.Lemma 9 (see [26]).
Letω
∈
A
∞ and b
∈
B
M
O
(
ℝ
n
). Then, for any cube Q
⊂
ℝ
n,
(33)
1
ω
(
Q
)
∫
Q
‍
exp
⁡
(
|
b
(
x
)
-
b
Q
|
C
0
∥
b
∥
*
)
ω
(
x
)
d
x
≤
C
,
∥
b
-
b
Q
∥
exp
⁡
L
,
Q
,
ω
≤
C
∥
b
∥
*
,
where C
0 and C are positive constants independent of b and Q, and ∥
b
∥
* is the B
M
O-norm of b.Lemma 10 (see [28]).
Let1
≤
p
<
∞, ω
p
∈
A
1, b
j
∈
B
M
O
(
ℝ
n
)
(
j
=
1
,
…
,
m
), and Q be a cube. Then for any positive integer m and k
=
0,1
,
…,
(34)
(
1
|
2
k
Q
|
∫
2
k
Q
‍
ω
p
(
x
)
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
|
p
d
x
)
1
/
p
≤
C
∥
b
→
∥
*
(
k
+
1
)
m
ess
inf
⁡
y
∈
Q
⁡
ω
(
y
)
.


## 3. Proof of Theorem6: The Case m
=
1

Whenm
=
1, we write b
=
b
1 and T
b
=
T
b
→ for simplicity. We need to prove that, for ω
∈
A
1 and b
∈
B
M
O
(
ℝ
n
), there exists constant C
>
0 such that, for all λ
>
0,
(35)
ω
(
{
x
∈
ℝ
n
:
|
T
b
f
(
x
)
|
>
λ
}
)
≤
C
∫
ℝ
n
‍
|
f
(
y
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
.For any fixedλ
>
0, we consider the Calderón-Zygmund decomposition of f at height λ and get a sequence of nonoverlapping cubes {
Q
i
}, where Q
i
=
Q
(
y
i
,
r
i
) is a cube centered at y
i with radius r
i, such that
(36)
|
f
(
x
)
|
≤
λ
,
for
a
.
e
.
x
∈
ℝ
n
∖
∪
i
Q
i
,
(37)
λ
<
1
|
Q
i
|
∫
Q
i
‍
|
f
(
x
)
|
d
x
≤
2
n
λ
,
i
=
1,2
,
…
.Denote byf
|
Q
i the restriction of f to Q
i. Let g
i
(
x
) be the projection of f
|
Q
i onto Y
i
=
span
⁡
{
ϕ
1
(
·
-
y
i
)
,
ϕ
2
(
·
-
y
i
)
,
…
,
ϕ
ℓ
(
·
-
y
i
)
}. We decompose f into two parts, f
=
g
+
h, where
(38)
g
(
x
)
=
{
f
(
x
)
,
x
∈
ℝ
n
∖
∪
i
Q
i
,
g
i
(
x
)
,
x
∈
Q
i
,
i
=
1,2
,
…
,
and h
(
x
)
=
f
(
x
)
-
g
(
x
)
=
∑
i
‍
h
i
(
x
) with h
i
(
x
)
=
f
(
x
)
-
g
i
(
x
) for x
∈
Q
i.Obviously,h
i is supported on Q
i and it follows from (18) that, for any 1
≤
k
≤
ℓ and any i (also see [2] p.170 or [3] (3.13)),
(39)
∫
Q
i
‍
ϕ
k
(
x
-
y
i
)
h
i
(
x
)
d
x
=
0
.Furthermore, we have(40)
|
g
(
x
)
|
≤
C
λ
,
a
.
e
.
x
∈
ℝ
n
.
Indeed, by (36) and (38) we have |
g
(
x
)
|
≤
λ, for a.e. x
∈
ℝ
n
∖
∪
i
Q
i. On the other hand, for any x
∈
∪
i
Q
i there exists an i so that x
∈
Q
i, and noting that g
i
(
x
) is the projection of f
|
Q
i onto Y
i, then it follows from Lemma 7 and (37) that
(41)
|
g
(
x
)
|
=
|
g
i
(
x
)
|
≤
sup
⁡
y
∈
Q
i
|
g
i
(
y
)
|
≤
C
|
Q
i
|
∫
Q
i
‍
|
f
(
y
)
|
d
y
≤
C
λ
.
So, (40) is verified.Sinceω
∈
A
1, then by (38), (41), and (16), we have
(42)
∫
ℝ
n
‍
|
g
(
x
)
|
ω
(
x
)
d
x
≤
∫
ℝ
n
∖
∪
i
Q
i
‍
|
f
(
x
)
|
ω
(
x
)
d
x
+
∫
∪
i
Q
i
‍
|
g
i
(
x
)
|
ω
(
x
)
d
x
≤
∫
ℝ
n
‍
|
f
(
x
)
|
ω
(
x
)
d
x
+
∑
i
‍
∫
Q
i
‍
(
C
|
Q
i
|
∫
Q
i
‍
|
f
(
y
)
|
d
y
)
ω
(
x
)
d
x
≤
∫
ℝ
n
‍
|
f
(
x
)
|
ω
(
x
)
d
x
+
C
∑
i
‍
ω
(
Q
i
)
|
Q
i
|
∫
Q
i
‍
|
f
(
y
)
|
d
y
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
ω
(
x
)
d
x
+
C
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
(
ess
inf
⁡
x
∈
Q
i
⁡
ω
(
x
)
)
d
y
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
ω
(
x
)
d
x
.For any cubeQ
i, by (16) and (37) we have
(43)
ω
(
Q
i
)
≤
C
|
Q
i
|
ess
inf
⁡
y
∈
Q
i
⁡
ω
(
y
)
≤
C
λ
-
1
∫
Q
i
‍
|
f
(
x
)
|
(
ess
inf
⁡
y
∈
Q
i
⁡
ω
(
y
)
)
d
x
≤
C
λ
-
1
∫
Q
i
‍
|
f
(
x
)
|
ω
(
x
)
d
x
.SetQ
i
*
=
2
n
Q
i and Ω
=
∪
i
Q
i
*; then
(44)
ω
(
Ω
)
≤
∑
i
‍
ω
(
Q
i
*
)
≤
C
∑
i
‍
ω
(
Q
i
)
≤
C
λ
-
1
∥
f
∥
L
1
(
ω
)
.
Thus
(45)
ω
(
{
x
∈
ℝ
n
:
|
T
b
f
(
x
)
|
>
λ
}
)
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
f
(
x
)
|
>
λ
2
}
)
+
ω
(
Ω
)
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
g
(
x
)
|
>
λ
2
}
)
+
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
h
(
x
)
|
>
λ
2
}
)
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
=
I
+
J
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
.For anyp
>
1, since ω
∈
A
1
⊂
A
p, then by Theorem 5, (40), and (42), we have
(46)
I
≤
C
λ
-
p
∫
ℝ
n
‍
|
T
b
g
(
x
)
|
p
ω
(
x
)
d
x
≤
C
λ
-
p
∫
ℝ
n
‍
|
g
(
x
)
|
p
ω
(
x
)
d
x
≤
C
λ
-
1
∫
ℝ
n
‍
|
g
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
-
1
∫
ℝ
n
‍
|
f
(
x
)
|
ω
(
x
)
d
x
.For the second termJ, since
(47)
T
b
h
(
x
)
=
∑
i
‍
T
b
h
i
(
x
)
=
∑
i
‍
(
b
(
x
)
-
b
Q
i
)
T
h
i
(
x
)

-

∑
i
‍
T
(
(
b
(
x
)

-

b
Q
i
)
h
i
)
(
x
)
,
then
(48)
J
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
∑
i
‍
(
b
(
x
)
-
b
Q
i
)
T
h
i
(
x
)
|
>
λ
4
}
)
+
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
∑
i
‍
T
(
(
b
(
x
)
-
b
Q
i
)
h
i
)
(
x
)
|
>
λ
4
}
)
=
J
(
1
)
+
J
(
2
)
.Let us considerJ
(
1
) first. Applying (39), condition (K
4), and Lemma 10, we have
(49)
J
(
1
)
≤
C
λ
∫
ℝ
n
∖
Ω
‍
|
∑
i
‍
(
b
(
x
)
-
b
Q
i
)
T
h
i
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
ℝ
n
∖
Q
i
*
‍
|
b
(
x
)
-
b
Q
i
|
×
|
∫
Q
i
‍
K
(
x
-
y
)
h
i
(
y
)
d
y
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
ℝ
n
∖
Q
i
*
‍
|
b
(
x
)
-
b
Q
i
|
×
(
∫
Q
i
‍
|
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
-
y
i
)
ϕ
k
(
y
-
y
i
)
|
×
|
h
i
(
y
)
|
d
y
∑
k
=
1
ℓ
‍
)
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
×
(
∫
ℝ
n
∖
Q
i
*
‍
|
∑
k
=
1
ℓ
‍
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
-
y
i
)
ϕ
k
(
y
-
y
i
)
|
×
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
∑
k
=
1
ℓ
‍
)
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
(
∫
|
x
-
y
i
|
>
2
|
y
-
y
i
|
‍
|
y
-
y
i
|
γ
|
x
-
y
|
n
+
γ
×
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
∫
|
x
-
y
i
|
>
2
|
y
-
y
i
|
‍
)
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
×
(
∑
s
=
1
∞
‍
∫
2
s
|
y
-
y
i
|
<
|
x
-
y
i
|
≤
2
s
+
1
|
y
-
y
i
|
‍
|
y
-
y
i
|
γ
|
x
-
y
i
|
n
+
γ
×
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
∑
s
=
1
∞
‍
)
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
(
∑
s
=
1
∞
‍
1
2
s
γ
(
2
s
+
1
|
y
-
y
i
|
)
n
×
∫
2
s
+
2
Q
i
‍
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x

∑
s
=
1
∞
‍
)
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
(
∑
s
=
1
∞
‍
1
2
s
γ
∥
b
∥
*
(
s
+
3
)
ess
inf
⁡
x
∈
Q
i
⁡
ω
(
x
)
)
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
ω
(
y
)
d
y
.
It follows from (42) that
(50)
J
(
1
)
≤
C
λ
∫
ℝ
n
‍
(
|
f
(
y
)
|
+
|
g
(
y
)
|
)
ω
(
y
)
d
y
≤
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.Now, let us considerJ
(
2
). By the weak type (
1,1
) estimate of T (see Theorem 3), (27), (41), and Lemmas 9 and 10, we have
(51)
J
(
2
)
≤
ω
(
{
x
∈
ℝ
n
:
|
T
(
∑
i
‍
(
b
-
b
Q
i
)
h
i
)
(
x
)
|
>
λ
4
}
)
≤
C
λ
∫
ℝ
n
‍
|
∑
i
‍
(
b
(
x
)
-
b
Q
i
)
h
i
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
b
(
x
)
-
b
Q
i
|
|
h
i
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
b
(
x
)
-
b
Q
i
|
|
f
(
x
)
-
g
i
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
x
)
|
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
+
C
λ
∑
i
‍
∫
Q
i
‍
|
g
i
(
x
)
|
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
log
⁡
L
,
Q
i
,
ω
∥
b
-
b
Q
i
∥
exp
⁡
L
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
(
1
|
Q
i
|
∫
Q
i
‍
|
f
(
y
)
|
d
y
)
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
log
⁡
L
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
(
1
|
Q
i
|
∫
Q
i
‍
|
b
(
x
)
-
b
Q
i
|
ω
(
x
)
d
x
)
d
y
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
log
⁡
L
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
(
ess
inf
⁡
x
∈
Q
i
⁡
ω
(
x
)
)
d
y
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
log
⁡
L
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.Note that (26) implies
(52)
∥
f
∥
L
log
⁡
L
,
Q
i
,
ω
≤
C
{
λ
+
λ
ω
(
Q
i
)
∫
Q
i
‍
Φ
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
}
.
Then by (43) we have
(53)
J
(
2
)
≤
C
∑
i
‍
{
ω
(
Q
i
)
+
∫
Q
i
‍
Φ
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
}
+
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
Φ
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
+
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
.Combining the estimates forJ
(
1
) and J
(
2
), we have
(54)
J
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
.
This along with (45) and (46) gives (35), which is the desired result.


## 4. Proof of Theorem6: The General Case m
>
1

In this section, we will use an induction argument to prove Theorem6 for the general case. To this end, we first introduce some notation.As in [22], given positive integers m and j
(
1
≤
j
≤
m
), we denote by 𝒞
j
m the family of all finite subsets σ
=
{
σ
(
1
)
,
σ
(
2
)
,
…
,
σ
(
j
)
} of {
1,2
,
…
,
m
} of j different elements. For any σ
∈
𝒞
j
m, we write σ
′
=
{
1,2
,
…
,
m
}
∖
σ.Forb
→
=
(
b
1
,
…
,
b
m
) with b
j
∈
B
M
O
(
ℝ
n
) and σ
=
{
σ
(
1
)
,
σ
(
2
)
,
…
,
σ
(
j
)
}
∈
𝒞
j
m
(
1
≤
j
≤
m
), we denote by b
→
σ
=
(
b
σ
(
1
)
,
b
σ
(
2
)
,
…
,
b
σ
(
j
)
), b
→
σ
′
=
(
b
σ
′
(
1
)
,
…
,
b
σ
′
(
m
-
j
)
), and ∥
b
→
∥
*
=
∥
b
1
∥
*
⋯
∥
b
m
∥
*, ∥
b
→
σ
∥
*
=
∥
b
σ
(
1
)
∥
*
⋯
∥
b
σ
(
j
)
∥
*. Write
(55)
(
b
→
(
x
)
-
b
→
(
y
)
)
σ
=
∏
i
=
1
j
‍
(
b
σ
(
i
)
(
x
)
-
b
σ
(
i
)
(
y
)
)
,
(
b
→
(
y
)
-
b
→
Q
)
σ
=
∏
i
=
1
j
‍
(
b
σ
(
i
)
(
y
)
-
(
b
σ
(
i
)
)
Q
)
,
where Q is a cube in ℝ
n and b
→
Q
=
(
(
b
1
)
Q
,
…
,
(
b
m
)
Q
). We also need the following notation:
(56)
T
b
→
σ
f
(
x
)
=
∫
ℝ
n
‍
(
b
→
(
x
)
-
b
→
(
y
)
)
σ
K
(
x
,
y
)
f
(
y
)
d
y
.Proof of Theorem6 (the general case m
>
1).
We have proved that Theorem6 is true for m
=
1 in Section 3. Now, we assume that Theorem 6 holds for all positive integer j
<
m; namely, for all 1
≤
j
<
m and any σ
∈
𝒞
j
m, we have
(57)
ω
(
{
x

∈

ℝ
n
:
|
T
b
→
σ
f
(
x
)
|

>

λ
}
)

≤

C
∫
ℝ
n
‍
Φ
j
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
.
For any fixedλ
>
0, we consider the Calderón-Zygmund decomposition of f at height λ as in Section 3 and use the notations {
Q
i
}, Q
i
*, g, h, h
i, and Ω as there.
For the same reason as in (45), we have
(58)
ω
(
{
x
∈
ℝ
n
:
|
T
b
→
f
(
x
)
|
>
λ
}
)
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
→
f
(
x
)
|
>
λ
}
)
+
ω
(
Ω
)
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
→
g
(
x
)
|
>
λ
2
}
)
+
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
→
h
(
x
)
|
>
λ
2
}
)
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
:
=
I
+
J
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
.
Similar to (46), we have
(59)
I
≤
C
λ
-
p
∫
ℝ
n
‍
|
T
b
→
g
(
x
)
|
p
ω
(
x
)
d
x
≤
C
λ
-
p
∫
ℝ
n
‍
|
g
(
x
)
|
p
ω
(
x
)
d
x
≤
C
λ
-
1
∥
f
∥
L
1
(
ω
)
.
Then
(60)
ω
(
{
x
∈
ℝ
n
:
|
T
b
→
f
(
x
)
|
>
λ
}
)
≤
J
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
.
Reasoning as the proof of Lemma 3.1 in [22] (pp. 683-684), we have
(61)
T
b
→
h
i
(
x
)
=
(
b
1
(
x
)
-
(
b
1
)
Q
i
)
⋯
(
b
m
(
x
)
-
(
b
m
)
Q
i
)
T
h
i
(
x
)
+
(
-
1
)
m
T
(
(
b
1
-
(
b
1
)
Q
i
)
⋯
(
b
m
-
(
b
m
)
Q
i
)
h
i
)
(
x
)
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
(
-
1
)
m
-
j
∫
ℝ
n
‍
(
b
→
(
x
)

-

b
→
Q
i
)
σ
(
b
→
(
y
)

-

b
→
Q
i
)
σ
′
×
K
(
x
,
y
)
h
i
(
y
)
d
y
.
Note that(62)
(
b
→
(
x
)
-
b
→
Q
i
)
σ
=
∏
s
=
1
j
‍
[
(
b
σ
(
s
)
(
x
)
-
b
σ
(
s
)
(
y
)
)
+
(
b
σ
(
s
)
(
y
)
-
(
b
σ
(
s
)
)
Q
i
)
]
,
(
b
→
(
y
)
-
b
→
Q
i
)
σ
′
=
∏
s
=
1
m
-
j
‍
[
b
σ
′
(
s
)
(
y
)
-
(
b
σ
′
(
s
)
)
Q
i
]
,
and expanding (
b
→
(
x
)
-
b
→
Q
i
)
σ
(
b
→
(
y
)
-
b
→
Q
i
)
σ
′, it is not difficult to check that
(63)
T
b
→
h
i
(
x
)
=
(
b
1
(
x
)
-
(
b
1
)
Q
i
)
⋯
(
b
m
(
x
)
-
(
b
m
)
Q
i
)
T
h
i
(
x
)
+
C
m
T
(
(
b
1
-
(
b
1
)
Q
i
)
⋯
(
b
m
-
(
b
m
)
Q
i
)
h
i
)
(
x
)
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
C
m
,
j
T
b
→
σ
(
(
b
→
-
b
→
Q
i
)
σ
′
h
i
)
(
x
)
.
This gives
(64)
|
T
b
→
h
(
x
)
|
=
|
∑
i
‍
T
b
→
h
i
(
x
)
|
≤
∑
i
‍
[
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
]
|
T
h
i
(
x
)
|
+
C
|
T
(
∑
i
‍
[
∏
j
=
1
m
‍
(
b
j
-
(
b
j
)
Q
i
)
]
h
i
)
(
x
)
|
+
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
|
T
b
→
σ
(
∑
i
‍
(
b
→
-
b
→
Q
i
)
σ
′
h
i
)
(
x
)
|
.
Thus,(65)
J
=
ω
(
{
x
∈
ℝ
n
∖
Ω
:
|
T
b
→
h
(
x
)
|
>
λ
2
}
)
≤
ω
(
{
x
∈
ℝ
n
∖
Ω
:
∑
i
‍
[
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
]
×
|
T
h
i
(
x
)
|
>
λ
6
∏
j
=
1
m
‍
}
)
+
ω
(
{
∏
j
=
1
m
‍
x
∈
ℝ
n
∖
Ω
:
C
×
|
T
(
∑
i
‍
[
∏
j
=
1
m
‍
(
b
j
-
(
b
j
)
Q
i
)
]
h
i
)
(
x
)
|
>
λ
6
}
)
+
ω
(
{
∑
j
=
1
m
-
1
‍
x
∈
ℝ
n
∖
Ω
:
C
×
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
|
T
b
→
σ
(
∑
i
‍
(
b
→
-
b
→
Q
i
)
σ
′
h
i
)
(
x
)
|
>
λ
6
}
)
:
=
J
1
+
J
2
+
J
3
.
Applying (39), condition (
K
4
), and Lemma 10, similar to the estimate of J
(
1
) in Section 3, we have
(66)
J
1
≤
C
λ
∑
i
‍
∫
ℝ
n
∖
Ω
‍
[
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
]
|
T
h
i
(
x
)
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
ℝ
n
∖
Q
i
*
‍
[
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
]
×
{
∫
Q
i
‍
|
K
(
x
-
y
)
-
∑
k
=
1
ℓ
‍
B
k
(
x
-
y
i
)
ϕ
k
(
y
-
y
i
)
|
×
|
h
i
(
y
)
|
d
y
∑
k
=
1
ℓ
‍
}
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
{
∏
j
=
1
m
‍
∫
|
x
-
y
i
|
>
2
|
y
-
y
i
|
‍
|
y
-
y
i
|
γ
|
x
-
y
|
n
+
γ
×
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
}
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
∑
s
=
1
∞
‍
{
∏
j
=
1
m
‍
∫
2
s
|
y
-
y
i
|
<
|
x
-
y
i
|
≤
2
s
+
1
|
y
-
y
i
|
‍
|
y
-
y
i
|
γ
|
x
-
y
i
|
n
+
γ
×
∏
j
=
1
m
‍
|
b
j
(
x
)

-

(
b
j
)
Q
i
|

ω
(
x
)
d
x

}

d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
∑
s
=
1
∞
‍
1
2
s
γ
1
(
2
s
+
1
|
y
-
y
i
|
)
n
×
∫
2
s
+
2
Q
i
‍
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
×
ω
(
x
)
d
x
d
y
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
h
i
(
y
)
|
∑
s
=
1
∞
‍
(
s
+
3
)
m
2
s
γ
∥
b
→
∥
*
ess
inf
⁡
y
∈
Q
i
⁡
ω
(
y
)
d
y
≤
C
λ
∥
f
∥
L
1
(
ω
)
.
ForJ
2, by the weak type (
1,1
) estimate for T (see Theorem 3), (27), (41), and Lemmas 9 and 10, similar to the estimate of J
(
2
) in Section 3, we have
(67)
J
2
≤
C
λ
∫
ℝ
n
‍
∑
i
‍
|
h
i
(
x
)
|
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
x
)
|
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
+
C
λ
∑
i
‍
∫
Q
i
‍
|
g
i
(
x
)
|
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
i
,
ω
∏
j
=
1
m
‍
∥
b
j
-
(
b
j
)
Q
i
∥
exp
⁡
L
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
(
1
|
Q
i
|
∫
Q
i
‍
|
f
(
y
)
|
d
y
)
×
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
×
(
1
|
Q
i
|
∫
Q
i
‍
∏
j
=
1
m
‍
|
b
j
(
x
)
-
(
b
j
)
Q
i
|
ω
(
x
)
d
x
)
d
y
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
(
ess
inf
⁡
y
∈
Q
i
⁡
ω
(
y
)
)
d
y
≤
C
λ
∑
i
‍
ω
(
Q
i
)
∥
f
∥
L
(
log
⁡
L
)
m
,
Q
i
,
ω
+
C
λ
∑
i
‍
∫
Q
i
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.
Then by (26) and (43) we have
(68)
J
2
≤
C
∑
i
‍
{
ω
(
Q
i
)
+
∫
Q
i
‍
Φ
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
}
+
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
Φ
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
+
C
λ
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
m
ω
(
y
)
d
y
.
Now, let us considerJ
3 by applying the induction hypothesis.
Noting thath
i
(
x
)
=
(
f
(
x
)
-
g
i
(
x
)
)
χ
Q
i
(
x
)  (
i
=
1,2
,
…
), we can split J
3 into two parts
(69)
J
3
≤
ω
(
{
λ
12
x
∈
ℝ
n
∖
E
:
C
×
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
|
T
b
→
σ
(
∑
i
‍
(
b
→
-
b
→
Q
i
)
σ
′
f
χ
Q
i
)
(
x
)
|
>
λ
12
}
)
+
ω
(
{
λ
12
x
∈
ℝ
n
∖
E
:
C
×
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
|
T
b
→
σ
(
∑
i
‍
(
b
→
-
b
→
Q
i
)
σ
′
g
i
χ
Q
i
)
(
x
)
|
>
λ
12
}
)
:
=
J
3
(
1
)
+
J
3
(
2
)
.
Forσ
∈
𝒞
j
m, we denote by σ
′
=
{
σ
′
(
1
)
,
σ
′
(
2
)
,
…
σ
′
(
m
-
j
)
}, so that
(70)
|
(
b
→
-
b
→
Q
i
)
σ
′
|
=
|
b
σ
′
(
1
)
-
(
b
σ
′
(
1
)
)
Q
i
|
⋯
|
b
σ
′
(
m
-
j
)
-
(
b
σ
′
(
m
-
j
)
)
Q
i
|
.
From Lemma9, there exist constants C
s
,
0 and C
s such that for s
=
1
⋯
m
-
j
(71)
1
ω
(
Q
i
)
∫
Q
i
‍
exp
⁡
(
|
b
σ
′
(
s
)
(
x
)
-
(
b
σ
′
(
s
)
)
Q
i
|
C
s
,
0
∥
b
σ
′
(
s
)
∥
*
)
ω
(
x
)
d
x
≤
C
s
.
Setγ
s
=
(
C
s
,
0
∥
b
σ
′
(
s
)
∥
*
)
-
1
(
s
=
1
,
…
m
-
j
); then it follows from the induction hypothesis and (31) that
(72)
J
3
(
1
)
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∫
ℝ
n
‍
Φ
j
(
∑
i
‍
|
f
(
y
)
|
λ
×
∑
i
‍
|
(
b
→

-

b
→
Q
i
)
σ
′
|
χ
Q
i
(
y
)
)
ω
(
y
)
d
y
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
Φ
j
(
|
f
(
y
)
|
λ
|
(
b
→
-
b
→
Q
i
)
σ
′
|
)
ω
(
y
)
d
y
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
Φ
m
(
|
f
(
y
)
|
γ
1
⋯
γ
m
-
j
·
λ
)
ω
(
y
)
d
y
+
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
{
∑
s
=
1
m
-
j
‍
∫
Q
i
‍
Ψ
(
γ
s
|
b
σ
′
(
s
)
-
(
b
σ
′
(
s
)
)
Q
i
|
)
×
ω
(
y
)
d
y
∑
s
=
1
m
-
j
‍
}
.
By (71) and (43), we have
(73)
∑
i
‍
{
∑
s
=
1
m
-
j
‍
∫
Q
i
‍
Ψ
(
γ
s
|
b
σ
′
(
s
)
-
(
b
σ
′
(
s
)
)
Q
i
|
)
ω
(
y
)
d
y
}
=
∑
i
‍
{
∑
s
=
1
m
-
j
‍
∫
Q
i
‍
[
exp
⁡
(
|
b
σ
′
(
s
)
(
x
)
-
(
b
σ
′
(
s
)
)
Q
i
|
C
ℓ
,
0
∥
b
∥
*
)
-
1
]
×
ω
(
y
)
d
y
∑
s
=
1
m
-
j
‍
[
exp
⁡
(
|
b
σ
′
(
s
)
(
x
)
-
(
b
σ
′
(
s
)
)
Q
i
|
C
ℓ
,
0
∥
b
∥
*
)
-
1
]
}
≤
∑
i
‍
∑
s
=
1
m
-
j
‍
C
s
ω
(
Q
i
)
≤
C
λ
-
1
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.
Noting thatΦ
m
(
a
b
)
≤
C
Φ
m
(
a
)
Φ
m
(
b
) for a
,
b
>
0, we have
(74)
J
3
(
1
)
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∫
ℝ
n
‍
Φ
m
(
|
f
(
y
)
|
λ
)
×
Φ
m
(
1
γ
1
⋯
γ
m
-
j
)
ω
(
y
)
d
y
+
C
λ
-
1
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
Φ
m
(
|
f
(
y
)
|
λ
)
ω
(
y
)
d
y
.
Finally, we considerJ
3
(
2
). By Jensen's inequality,
(75)
Φ
m
(
|
f
Q
i
|
λ
)
≤
Φ
m
(
1
|
Q
i
|
∫
Q
i
‍
|
f
(
x
)
|
λ
d
x
)
≤
1
|
Q
i
|
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
d
x
.
By the induction hypothesis, (31), and (75), similar to the estimate of J
3
(
1
), we have
(76)
J
3
(
2
)
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∫
ℝ
n
‍
Φ
j
(
|
f
Q
i
|
λ
∑
i
‍
|
(
b
→
-
b
→
Q
i
)
σ
′
|
χ
Q
i
(
y
)
)
×
ω
(
y
)
d
y
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
Φ
j
(
|
f
Q
i
|
λ
|
(
b
→
-
b
→
Q
i
)
σ
′
|
)
ω
(
y
)
d
y
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
Φ
m
(
|
f
Q
i
|
γ
1
⋯
γ
m
-
j
·
λ
)
ω
(
y
)
d
y
+
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
{
∑
s
=
1
m
-
j
‍
∫
Q
i
‍
Ψ
(
γ
s
|
b
σ
′
(
s
)
-
(
b
σ
′
(
s
)
)
Q
i
|
)
×
ω
(
y
)
d
y
∑
s
=
1
m
-
j
‍
∑
s
=
1
m
-
j
‍
}
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
{
1
|
Q
i
|
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
d
x
}
ω
(
y
)
d
y
+
C
λ
-
1
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
.
Applying (16), we have
(77)
∫
Q
i
‍
{
1
|
Q
i
|
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
d
x
}
ω
(
y
)
d
y
=
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
{
1
|
Q
i
|
∫
Q
i
‍
ω
(
y
)
d
y
}
d
x
≤
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
ess
inf
⁡
y
∈
Q
i
⁡
ω
(
y
)
d
x
≤
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
ω
(
x
)
d
x
.
Then,
(78)
J
3
(
2
)
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
𝒞
j
m
‍
∑
i
‍
∫
Q
i
‍
Φ
m
(
|
f
(
x
)
|
λ
)
ω
(
x
)
d
x
+
C
λ
-
1
∫
ℝ
n
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
∫
ℝ
n
‍
Φ
m
(
|
f
(
y
)
|
λ
)
d
y
.
This along with (69) and (74) gives
(79)
J
3
≤
C
∫
ℝ
n
‍
Φ
m
(
|
f
(
x
)
|
λ
)
d
x
.
By (60), (65), and the above estimates for J
1, J
2, and J
3, we obtain
(80)
ω
(
{
x
∈
ℝ
n
:
|
T
b
→
f
(
x
)
|
>
λ
}
)
≤
C
∫
ℝ
n
‍
Φ
m
(
|
f
(
x
)
|
λ
)
d
x
+
C
λ
-
1
∥
f
∥
L
1
(
ω
)
≤
C
∫
ℝ
n
‍
|
f
(
y
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
m
ω
(
y
)
d
y
.
The proof of the general case of Theorem6 is therefore completed.



---

*Source: 641705-2014-05-04.xml*