# Poincaré-Type Inequalities for the Composite Operator inL

Authors: Guannan Shi; Yuming Xing; Baiqing Sun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first establish the local Poincaré inequality withL
𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous A-harmonic equation. Then, according to the definition of L
𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality with L
𝒜-averaging domains. Finally, we give some illustrations for these theorems.


---

## Body

## 1. Introduction

Poincaré inequality applied to differential forms has a vital role in PDEs, nonlinear analysis, and other related fields. With the further research conducted, we have established various versions of Poincaré inequality under different conditions. From [1–8], we have obtained the Poincaré inequality for the solution to the A-harmonic equation in uniformly bounded domain, John domains, and L
s-averaging domains. Nevertheless, most of these Poincaré inequalities are developed in L
s-averaging domains. In this paper, we will establish the Poincaré inequality for the composition of the sharp maximal operator and potential operator in L
𝒜-averaging domains. As we all know, both the uniformly bounded domain and John domains are special L
s-averaging domains, and the L
s-averaging domains are also particular L
𝒜-averaging domains, so the following results are the generalizations of the Poincaré inequality in L
s-averaging domains.For convenience, we firstly introduce some notations and terminologies. Except for special instructions,E
⊆
ℝ
m is a bounded domain, |
E
| denotes the Lebesgue measure of E, and m
≥
2. The constant K and C can be varied at each step of the proof. Suppose that B
x
r is a ball, with a radius r, centered at x. For any ρ
>
0, B
⊆
E and ρ
B
⊆
E have the same center and satisfy diam
⁡
(
ρ
B
)
=
ρ
diam
⁡
(
B
). Let Λ
l
(
ℝ
m
) be the space of all l-forms in ℝ
m, which is expanded by the exterior product of e
ℬ
=
e
i
1
∧
e
i
2
∧
⋯
∧
e
i
l, where ℬ
=
(
i
1
,
…
,
i
l
)
,
1
≤
i
1
<
⋯
<
i
l
≤
m, l
=
1,2
,
…
,
m. C
∞
(
Λ
l
E
) is the space of a smooth l-form on E. We use D
′
(
E
,
Λ
l
) to denote the space of all differential l-forms on E; that is, w
(
x
) belongs to D
′
(
E
,
Λ
l
) if and only if there exist some lth-differential functions w
ℬ in E such that w
(
x
)
=
∑
ℬ
‍
w
ℬ
(
x
)
d
x
ℬ
=
∑
‍
w
i
1
i
2
⋯
i
l
(
x
)
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
l. L
p
(
E
,
Λ
l
) is a Banach space with the norm equipped by ∥
w
(
x
)
∥
p
,
E
=
(
∫
E
‍
|
w
(
x
)
|
p
d
x
)
1
/
p, where w
(
x
)
∈
D
′
(
E
,
Λ
l
) and every coefficient function w
ℬ
∈
L
p
(
E
)
,
0
<
p
<
∞. In fact, w
(
x
) on E is the Schwartz distribution. If ω
(
x
)
>
0 a.e. and ω
(
x
)
∈
L
loc
⁡
1
(
ℝ
m
), ω
(
x
) is called a weight. Let d
μ
=
ω
(
x
)
d
x; then L
p
(
E
,
Λ
l
,
ω
) is a weighted Banach space with the norm expressed by ∥
w
(
x
)
∥
p
,
E
,
ω
=
(
∫
E
‍
|
w
(
x
)
|
p
ω
(
x
)
d
x
)
1
/
p. In this notation, the exterior derivative is denoted by d and Hodge codifferential operator is expressed by d
⋆. Search [9] for more details.Considering our purpose, we intend to give a brief discussion about theA-harmonic equation for the differential form. The following equation is called a nonhomogeneous A-harmonic equation:
(1)
d
⋆
A
(
x
,
d
w
)
=
B
(
x
,
d
w
)
,
where A
:
E
×
∧
l
(
ℝ
m
)
→
∧
l
(
ℝ
m
) and B
:
E
×
∧
l
(
ℝ
m
)
→
∧
l
-
1
(
ℝ
m
) satisfy the conditions:
(2)
|
A
(
x
,
ξ
)
|
≤
a
|
ξ
|
p
-
1
,
A
(
x
,
ξ
)
·
ξ
≥
|
ξ
|
p
,
|
B
(
x
,
ξ
)
|
≤
b
|
ξ
|
p
-
1
,
for almost every x
∈
E and all ξ
∈
∧
l
(
ℝ
m
). Here, a
,
b
>
0 are constants and 1
<
p
<
∞ is a fixed exponent associated with (1). If B
=
0, the equation d
⋆
A
(
x
,
d
w
)
=
0 is called a homogenous A-harmonic equation. See [9] for more information.In order to describe it easily, we first give some definitions in this part.Definition 1.
LetE
⊆
ℝ
m be a bounded domain and w
(
x
)
∈
L
p
(
E
,
Λ
l
); the sharp maximal operator 𝕄
s
♯ is equipped with
(3)
𝕄
s
♯
(
w
)
=
𝕄
s
♯
w
=
𝕄
s
♯
w
(
x
)
=
sup
⁡
r
>
0
(
1
|
B
x
r
|
∫
B
x
r
‍
|
w
(
t
)
-
w
B
x
r
|
s
d
t
)
1
/
s
,
where B
x
r is the ball of radius r, centered at x, 1
≤
s
≤
p,  p
≥
1.
Especially, if we takes
=
1, denote 𝕄
s
♯
≜
𝕄
♯.Definition 2 (see [10]).
Suppose thatw
(
x
) is a differential l-form; the potential operator P is expressed by
(4)
P
w
(
x
)
=
∑
ℬ
‍
∫
E
‍
K
(
x
,
y
)
w
ℬ
(
y
)
d
y
d
x
β
,
where the nonnegative and measurable function K
(
x
,
y
), defined on the set {
(
x
,
y
)
∣
x
≠
y
,
x
,
y
∈
ℝ
m
}, is a kernel function, and the summation is over all ordered l-tuple ℬ.Definition 3.
Take an increasingly continuous function𝒜
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) as a convex function with 𝒜
(
0
)
=
0, and E
⊆
ℝ
m is a bounded domain, for any w
(
x
)
∈
L
p
(
E
); the Orlicz norm for differential form is denoted by
(5)
∥
w
∥
L
E
,
μ
𝒜
=
inf
⁡
{
λ
>
0
∣
1
μ
(
E
)
∫
E
‍
𝒜
(
λ
-
1
|
u
|
d
μ
)
<
1
}
,
where measure μ satisfies d
μ
=
ω
(
x
)
d
x
,
ω
(
x
) is a weight.
We call𝒜 an Orlicz function if 𝒜
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) is an increasingly continuous function and satisfies 𝒜
(
0
)
=
0 and 𝒜
(
∞
)
=
∞. Meanwhile, if the Orlicz function 𝒜
(
t
) is a convex function, it is called a Young function.
Based on the above definition, we get the notation ofL
𝒜-averaging domains.Definition 4 (see [3]).
Let𝒜 be a Young function; the proper domain E
⊆
ℝ
m is called the L
𝒜-averaging domains if μ
(
E
)
<
∞ and there exists a constant C
>
0 such that for any B
0
⊆
E and 𝒜
(
|
w
|
)
∈
L
loc
⁡
1
(
E
,
μ
), w satisfies
(6)
1
μ
(
E
)
∫
E
‍
𝒜
(
τ
|
w
-
w
B
0
|
)
d
μ
≤
C
sup
⁡
4
B
⊂
E
1
μ
(
B
)
∫
B
‍
𝒜
(
σ
|
w
-
w
B
|
)
d
μ
,
where the measure μ is denoted by d
μ
=
ω
(
x
)
d
x, ω
(
x
) is a weight, σ and τ are constants with 0
<
τ, σ
<
1, and the supremum is over all balls B
⊂
E with 4
B
⊂
E.
Notice that if we let𝒜
(
t
)
=
t
s, L
𝒜-averaging domains become the L
s-averaging domains, so L
𝒜-averaging domains are the generalization of L
s-averaging domains.Definition 5 (see [11]).
We callw
(
x
)
∈
D
′
(
E
,
Λ
l
) belongs to the WRH(
Λ
l
,
E
)-class, l
=
0,1
,
…
,
m, if for any constants 0
<
s, t
<
∞ and any ball B
⊂
E with ρ
B
⊂
E, there exists a constant C
>
0 such that w
(
x
) satisfies
(7)
∥
w
∥
s
,
B
≤
C
|
B
|
(
t
-
s
)
/
s
t
∥
w
∥
t
,
ρ
B
,
where ρ
>
1 is a constant.Remark 6.
Ifw
(
x
) is a solution to the A-harmonic equation, we can prove that w
(
x
) belongs to the WRH(
Λ
l
,
E
)-class.


## 2. Main Results

Before the main results are given, we need to impose some restrictions on the kernel functionK
(
x
,
y
) and Young function 𝒜. Firstly, let the kernel function satisfy the standard estimates; it is equal to say that if there exist 0
<
δ
<
1 and a constant c
>
0 such that for any point z
∈
{
z
:
|
x
-
z
|
<
(
1
/
2
)
|
x
-
y
|
,
x
,
y
∈
ℝ
m
}, the kernel function K
(
x
,
y
) satisfies that(1)
K
(
x
,
y
)
≤
c
|
x
-
y
|
-
m, x
≠
y;
(2)
|
K
(
x
,
y
)
-
K
(
z
,
y
)
|
≤
c
|
x
-
z
|
δ
|
x
-
y
|
-
m
-
δ, x
≠
y;
(3)
|
K
(
y
,
x
)
-
K
(
y
,
z
)
|
≤
c
|
x
-
z
|
δ
|
x
-
y
|
-
m
-
δ, x
≠
y, where function K
(
x
,
y
)
:
ℝ
m
×
ℝ
m
→
ℝ, m
≥
1.With regard to the Young function𝒜, we let the Young function 𝒜 belong to the G
(
p
,
q
,
C
)-class (
1
≤
p
<
q
<
∞
,
C
≥
1
); that is, for any t
>
0, the Young function 𝒜 satisfies that(1)
1
/
C
≤
𝒜
(
t
1
/
p
)
/
f
(
t
)
≤
C;
(2)
1
/
C
≤
𝒜
(
t
1
/
q
)
/
g
(
t
)
≤
C,where f and g are the increasingly convex and concave functions defined on [
0
,
∞
], respectively.Now, we establish these two important theorems based on the above conditions.Theorem 7.
Suppose that the Young function𝒜 belongs to the G
(
p
,
q
,
C
)-class, w
∈
C
∞
(
Λ
l
E
) is a solution to the nonhomogenous A-harmonic equation, the sharp maximal operator is noted by 𝕄
s
♯, P is the potential operator with its kernel function K
(
x
,
y
) satisfying the standard estimates, 1
≤
s
<
p
,
q
<
∞, and the bounded subset E
⊆
ℝ
m is the L
𝒜-averaging domains. Then, for any ball B
⊆
E, one gets
(8)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
L
B
𝒜
≤
K
∥
w
∥
L
ρ
B
𝒜
,
where B and ρ
B
⊆
E and the constant ρ
>
1.Based on the above theorem, we can establish the following theorem for the global Poincaré inequality inL
𝒜-averaging domains.Theorem 8.
Suppose that the Young function𝒜 belongs to the G
(
p
,
q
,
C
)-class, w
∈
C
∞
(
Λ
l
E
) is a solution to the nonhomogenous A-harmonic equation, the sharp maximal operator is denoted by 𝕄
s
♯, P is the potential operator with its kernel function K
(
x
,
y
) satisfying the standard estimates, 1
≤
s
<
p, q
<
∞, and the bounded subset E
⊆
ℝ
m is the L
𝒜-averaging domains. Then, one has
(9)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
0
∥
L
E
𝒜
≤
K
∥
w
∥
L
E
𝒜
,
where B
0
⊆
B is a fixed ball, which appears in Definition 4.


## 3. Preliminary Results

For proving the theorems in Section2, we will show and demonstrate some lemmas in this part.Lemma 9 (see [9]).
Let0
<
p, q
<
∞, and 1
/
t
=
(
1
/
p
)
+
(
1
/
q
), if f and g are the measurable functions defined on ℝ
m, then
(10)
∥
f
g
∥
t
,
I
≤
∥
f
∥
p
,
I
·
∥
g
∥
q
,
I
,
for any I
⊆
ℝ
m.Lemma 10 (see [5]).
LetP be the potential operator applied on a differential form with E
⊆
ℝ
m, w
(
x
)
∈
W
R
H
(
Λ
l
,
E
), and assume that the weight ω
(
x
) belongs to A
(
α
,
β
,
E
) with α
,
β
>
0. Then, there exists a constant C, independent of w
(
x
) such that
(11)
∥
P
(
w
)
-
(
P
(
w
)
)
B
∥
s
,
B
,
ω
≤
C
|
B
|
diam
⁡
(
B
)
∥
w
∥
s
,
B
,
ω
,
for any B
⊂
E, where s
>
1 is a constant.Remark 11.
If we takeω
(
x
)
≡
1, we get
(12)
∥
P
(
w
)
-
(
P
(
w
)
)
B
∥
s
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
w
∥
s
,
B
.Lemma 12 (see [3]).
TakeΨ defined on [
0
,
+
∞
) to be a strictly increasing convex function, Ψ
(
0
)
=
0, and E
⊂
ℝ
m is a domain. Assume that w
(
x
)
∈
D
′
(
E
,
Λ
l
) satisfies Ψ
(
|
w
|
)
∈
L
1
(
E
,
μ
) and, for any constant c,
(13)
μ
{
x
∈
E
:
|
w
-
c
|
>
0
}
>
0
,
where μ is a Radon measure defined by d
μ
(
x
)
=
ω
(
x
)
d
x with a weight ω
(
x
); then for any a
>
0, one obtains
(14)
∫
E
‍
Ψ
(
a
2
|
w
-
w
E
|
)
d
μ
≤
∫
E
‍
Ψ
(
a
|
w
|
)
d
μ
.Lemma 13.
Ifω
(
x
)
∈
A
r
(
E
), then there exist constants α
>
1 and K, not dependent on ω, such that
(15)
∥
ω
∥
α
,
B
≤
K
|
B
|
(
1
/
α
)
-
1
∥
ω
∥
1
,
B
,
for all balls B contained in E.Lemma 14.
The sharp maximal operator𝕄
s
♯ is denoted by Definition 1, and the potential operator P is defined by Definition 2 with the kernel function K
(
x
,
y
) satisfying the standard estimates, w
(
x
)
∈
L
t
(
E
,
Λ
l
)
∩
C
∞
(
Λ
l
E
)
(
l
=
1,2
,
…
,
m
), t
≥
1. Then, there exists a constant K
>
0, independent of w, such that
(16)
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
t
d
x
≤
K
|
B
|
1
+
(
1
/
m
)
∥
w
∥
t
,
B
,
for all balls B
⊂
E.Proof .
LetB be a ball in E, using Lemma 10 on any B
x
r
⊂
B, we have
(17)
(
1
|
B
x
r
|
∫
B
x
r
‍
|
P
(
w
)
-
(
P
(
w
)
)
B
x
r
|
s
d
x
)
1
/
s
≤
K
|
B
x
r
|
diam
⁡
(
B
x
r
)
|
B
x
r
|
-
1
/
s
∥
w
∥
s
,
B
x
r
≤
K
|
B
x
r
|
1
-
(
1
/
s
)
diam
⁡
(
B
x
r
)
∥
w
∥
s
,
B
≤
K
|
B
|
1
-
(
1
/
s
)
+
(
1
/
m
)
∥
w
∥
s
,
B
.
From Lemma 3.5 in [7], it follows that
(18)
∥
w
∥
s
,
B
≤
|
B
|
(
1
/
s
)
-
(
1
/
t
)
∥
w
∥
t
,
B
,
where 0
<
s
≤
t
<
∞. Substituting (18) into (17) yields
(19)
(
1
|
B
x
r
|
∫
B
x
r
‍
|
P
(
w
)
-
(
P
(
w
)
)
B
x
r
|
s
d
x
)
1
/
s
≤
K
|
B
|
1
-
(
1
/
t
)
+
(
1
/
m
)
∥
w
∥
t
,
B
.
Taking the supremum for r, we get that
(20)
sup
⁡
r
>
0
(
1
|
B
x
r
|
∫
B
x
r
‍
|
P
(
w
)
-
(
P
(
w
)
)
B
x
r
|
s
d
x
)
1
/
s
≤
sup
⁡
r
>
0
K
|
B
|
1
-
(
1
/
t
)
+
(
1
/
m
)
∥
w
∥
t
,
B
=
K
|
B
|
1
-
(
1
/
t
)
+
(
1
/
m
)
∥
w
∥
t
,
B
.
That is,
(21)
𝕄
s
♯
(
P
(
w
)
)
≤
K
|
B
|
1
-
(
1
/
t
)
+
(
1
/
m
)
∥
w
∥
t
,
B
.
According to the definition of L
t
(
E
) norm and formula (21), it yields
(22)
∥
𝕄
s
♯
(
P
(
w
)
)
∥
t
,
B
=
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
|
t
d
x
)
1
/
t
≤
(
∫
B
‍
|
K
|
B
|
1
-
(
1
/
t
)
+
(
1
/
m
)
∥
w
∥
t
,
B
|
t
d
x
)
1
/
t
=
K
|
B
|
1
+
(
1
/
m
)
∥
w
∥
t
,
B
.
Choosing Ψ
(
t
)
=
2
t, a
=
2, and ω
(
x
)
≡
1 in Lemma 12, we have
(23)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
t
,
B
=
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
t
d
x
)
1
/
t
≤
(
∫
B
‍
2
t
|
𝕄
s
♯
(
P
(
w
)
)
|
t
d
x
)
1
/
t
≤
K
|
B
|
1
+
(
1
/
m
)
∥
w
∥
t
,
B
.
The proof of Lemma 14 has been completed.Lemma 15.
Suppose thatw
(
x
)
∈
C
∞
(
Λ
l
E
) is a solution to the A-harmonic equation, E
⊂
ℝ
m is a bounded domain, P is a potential operator with the kernel function K
(
x
,
y
) satisfying the standard estimates, and the sharp maximal operator 𝕄
s
♯ is expressed by Definition 1, 1
≤
s
<
p, q
<
∞. Then, there exists a constant K
>
0, such that
(24)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
q
,
B
≤
K
∥
w
∥
p
,
ρ
B
,
where the ball B
⊂
E with ρ
B
⊂
E, constant ρ
>
1, the measure μ is defined by d
μ
=
ω
(
x
)
d
x, weight ω
(
x
)
∈
A
r
(
E
), ω
(
x
)
≥
δ
>
0, for some r
>
1 and a constant δ.Proof .
Because1
/
q
=
(
(
α
-
1
)
/
α
q
)
+
(
1
/
α
q
), for any B with ρ
B contained in E, using Lemmas 9 and 14, we have
(25)
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
μ
)
1
/
q
=
(
∫
B
‍
(
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
ω
1
/
q
)
q
d
x
)
1
/
q
≤
(
∫
B
‍
(
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
α
q
/
(
α
-
1
)
)
d
x
)
(
α
-
1
)
/
α
q
×
(
∫
B
‍
ω
α
d
x
)
1
/
q
α
≤
K
|
B
|
(
1
-
α
)
/
α
q
∥
ω
∥
1
,
B
1
/
q
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
q
α
/
(
α
-
1
)
,
B
.
According to Lemma 14 and Definition 5, letting p
=
r
×
z, we get
(26)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
α
q
/
(
α
-
1
)
,
B
≤
K
|
B
|
1
+
(
1
/
m
)
∥
w
∥
α
q
/
(
α
-
1
)
,
B
=
K
|
B
|
1
+
(
1
/
m
)
+
(
(
z
(
α
-
1
)
-
α
q
)
/
α
q
z
)
∥
w
∥
z
,
ρ
B
.
Therefore, we know that
(27)
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
μ
)
1
/
q
≤
K
|
B
|
(
1
-
α
)
/
q
α
∥
ω
∥
1
,
B
1
/
q
|
B
|
1
+
(
1
/
m
)
+
(
(
z
(
α
-
1
)
-
α
q
)
/
α
q
z
)
∥
w
∥
z
,
ρ
B
=
K
|
B
|
1
+
(
1
/
m
)
-
(
1
/
z
)
(
μ
(
B
)
)
1
/
q
∥
w
∥
z
,
ρ
B
.
Because of 1
/
z
=
(
1
/
p
)
+
(
(
p
-
z
)
/
z
p
), and using generalized Hölder’s inequality, we get
(28)
∥
w
∥
z
,
ρ
B
=
(
∫
ρ
B
‍
(
|
w
|
ω
1
/
p
ω
-
1
/
p
)
z
d
x
)
1
/
z
≤
(
∫
ρ
B
‍
|
w
|
p
ω
d
x
)
1
/
p
·
(
∫
ρ
B
‍
ω
1
/
(
r
-
1
)
d
x
)
(
r
-
1
)
/
p
=
∥
w
∥
p
,
ρ
B
,
ω
∥
ω
-
1
∥
1
/
(
r
-
1
)
,
ρ
B
1
/
p
.
In the light of ω
∈
A
r
(
E
), finding details in [9], we know
(29)
sup
⁡
B
⊂
E
(
1
|
B
|
∫
B
‍
ω
d
x
)
(
1
|
B
|
∫
B
‍
(
1
ω
)
1
/
(
r
-
1
)
d
x
)
1
/
(
r
-
1
)
<
∞
.
Therefore, we can see that
(30)
∥
1
ω
∥
1
/
(
r
-
1
)
,
ρ
B
1
/
p
=
∥
ω
∥
1
,
ρ
B
-
1
/
p
·
∥
ω
∥
1
,
ρ
B
1
/
p
·
∥
1
ω
∥
1
/
(
r
-
1
)
,
ρ
B
1
/
p
≤
K
(
μ
(
ρ
B
)
)
-
1
/
p
|
ρ
B
|
1
/
z
≤
K
(
μ
(
ρ
B
)
)
-
1
/
p
|
B
|
1
/
z
.
In addition, considering ω
≥
δ
>
0, so we have that
(31)
μ
(
ρ
B
)
=
∫
ρ
B
‍
d
μ
≥
∫
ρ
B
‍
δ
d
x
=
δ
|
ρ
B
|
.
Combining (27), (28), and (31), we obtain
(32)
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
μ
)
1
/
q
≤
K
|
ρ
B
|
1
+
(
1
/
m
)
-
(
1
/
p
)
(
μ
(
B
)
)
1
/
q
(
∫
ρ
B
‍
|
w
|
1
/
p
d
μ
)
1
/
q
≤
K
|
E
|
1
+
(
1
/
m
)
-
(
1
/
p
)
(
μ
(
E
)
)
1
/
q
(
∫
ρ
B
‍
|
w
|
1
/
p
d
μ
)
1
/
q
≤
K
∥
w
∥
p
,
ρ
B
.
Therefore, we finish the proof of this lemma.


## 4. Demonstration of Main Results

According to the above definitions and lemmas, we will prove these two theorems in detail. Firstly, let us prove Theorem7.Proof of Theorem7.
LetB and ρ
B
⊆
E, f and g are, respectively, convex and concave increasing function, use Lemma 15, and take ω
(
x
)
≡
1; then
(33)
𝒜
(
λ
-
1
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
x
)
)
1
/
q
≤
𝒜
(
λ
-
1
K
(
∫
ρ
B
‍
|
w
|
p
d
x
)
1
/
p
)
=
𝒜
(
(
λ
-
p
K
p
∫
ρ
B
‍
|
w
|
p
d
x
)
1
/
p
)
≤
C
f
(
λ
-
p
K
p
∫
ρ
B
‍
|
w
|
p
d
x
)
≤
C
∫
ρ
B
‍
f
(
λ
-
p
K
p
|
w
|
p
)
d
x
.
Because f
(
t
)
≤
C
𝒜
(
t
1
/
p
), we know that
(34)
∫
ρ
B
‍
f
(
λ
-
p
K
p
|
w
|
p
)
d
x
≤
C
∫
ρ
B
‍
𝒜
(
K
λ
-
1
|
w
|
)
d
x
.
Furthermore, we obtain
(35)
𝒜
(
λ
-
1
(
∫
B
‍
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
x
)
)
1
/
q
≤
C
∫
ρ
B
‍
𝒜
(
K
λ
-
1
|
w
|
)
d
x
.
For function g, using Jensen's inequality, we get
(36)
∫
B
‍
𝒜
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
d
x
≤
g
(
∫
B
‍
g
-
1
(
𝒜
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
)
d
x
)
≤
g
(
C
∫
B
‍
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
q
d
x
)
≤
C
𝒜
(
(
C
∫
B
‍
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
q
d
x
)
1
/
q
)
=
C
𝒜
(
λ
-
1
(
C
∫
B
‍
(
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
q
d
x
)
1
/
q
)
.
Using the doubling property of 𝒜 for the above the formula, we have
(37)
∫
B
‍
𝒜
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
d
x
≤
K
𝒜
(
λ
-
1
(
∫
B
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
‍
d
x
)
1
/
q
)
≤
K
∫
ρ
B
‍
𝒜
(
λ
-
1
|
w
|
)
d
x
.
The proof of Theorem 7 has been finished.Now, we will use Definition4 and Theorem 7 to prove Theorem 8.Proof of Theorem8.
According to Definition4, we can know
(38)
∫
E
‍
𝒜
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
0
|
)
d
x
≤
K
sup
⁡
B
⊆
E
∫
B
‍
𝒜
(
λ
-
1
|
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
d
x
≤
K
sup
⁡
B
⊆
E
∫
ρ
B
‍
𝒜
(
λ
-
1
|
w
|
)
d
x
≤
K
sup
⁡
B
⊆
E
∫
E
‍
𝒜
(
λ
-
1
|
w
|
)
d
x
.
Because sup
⁡
B
⊆
E
∫
E
‍
𝒜
(
λ
-
1
|
w
|
)
d
x is independent on the ball B, we obtain that
(39)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
0
∥
L
E
𝒜
≤
K
∥
w
∥
L
E
𝒜
.
We finish the proof of Theorem 8.


## 5. Applications

In this part, we firstly use Theorem8 to do an estimate for a solution to the Laplace equation Δ
u
=
0.Example 16.
Letu be a differential 2-form in ℝ
m, and
(40)
u
=
1
ϑ
(
x
1
d
x
2
∧
d
x
3
+
x
2
d
x
1
∧
d
x
3
+
x
3
d
x
1
∧
d
x
2
)
,
where ϑ
=
x
1
2
+
x
2
2
+
x
3
2. It is very easy to obtain that |
u
|
=
1 and d
u
=
0, so u is a solution for the Laplace equation Δ
u
=
0. If we take
(41)
𝒜
(
t
)
=
t
log
⁡
+
t
=
{
t
,
t
≤
e
t
log
⁡
t
,
t
>
e
,
then 𝒜
(
t
) is a Young function and belongs to the G
(
p
,
q
,
C
)-class, with 𝒜
(
|
u
|
)
∈
L
1
(
E
). According to Theorem 8, we get that, for any fixed B
0
⊂
E, there exists a constant K
>
0 such that
(42)
∥
𝕄
s
♯
(
P
(
u
)
)
-
(
𝕄
s
♯
(
P
(
u
)
)
)
B
0
∥
L
E
𝒜
≤
K
∥
1
∥
L
E
𝒜
,
where ∥
1
∥
L
E
𝒜
=
inf
⁡
{
λ
>
0
∣
(
1
/
|
E
|
)
∫
E
‍
𝒜
(
λ
-
1
log
⁡
+
λ
)
d
t
<
1
}.Now, our aim is to prove the following corollary by using Theorem7.Corollary 17.
Suppose that the Young function𝒜 belongs to the G
(
p
,
q
,
C
)-class, and w
∈
C
∞
(
Λ
l
E
) is a solution to the nonhomogenous A-harmonic equation. The sharp maximal operator is noted by M
s
♯, P is the potential operator with its kernel function K
(
x
,
y
) satisfying the standard estimates (1
≤
s
<
p, q
<
∞), and the bounded E
⊆
ℜ
m is the L
𝒜-averaging domains. Then, there exists a constant K
>
0, such that
(43)
∥
𝕄
s
♯
(
P
(
w
)
)
∥
L
B
𝒜
≤
K
∥
w
∥
L
ρ
B
𝒜
,
where B
⊂
E with ρ
B
⊆
E, and the constant ρ
>
1.Proof .
By using Minkowski inequality, we know that(44)
∥
𝕄
s
♯
(
P
(
w
)
)
∥
L
B
𝒜
≤
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
L
B
𝒜
+
∥
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
L
B
𝒜
.
From [12] and formula (22), we have
(45)
∫
B
‍
𝒜
(
λ
-
1
|
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
d
t
=
f
·
f
-
1
(
∫
B
‍
𝒜
(
λ
-
1
|
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
d
t
)
≤
f
(
K
∫
B
‍
f
-
1
(
𝒜
(
λ
-
1
|
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
)
)
d
t
)
≤
f
(
K
∫
B
‍
λ
-
q
|
(
𝕄
s
♯
(
P
(
w
)
)
)
B
|
q
d
t
)
≤
f
(
K
∫
B
‍
λ
-
q
|
(
𝕄
s
♯
(
P
(
w
)
)
)
|
q
d
t
)
≤
f
(
K
λ
-
q
∫
B
‍
|
w
|
q
d
t
)
≤
K
∫
B
‍
f
(
λ
-
q
|
w
|
q
)
d
t
≤
K
∫
B
‍
𝒜
(
λ
-
1
|
w
|
)
d
t
≤
K
∥
w
∥
L
ρ
B
𝒜
,
where ρ
>
1. In addition, according to Theorem 7, there exists a constant K such that
(46)
∥
𝕄
s
♯
(
P
(
w
)
)
-
(
𝕄
s
♯
(
P
(
w
)
)
)
B
∥
L
B
𝒜
≤
K
∥
w
∥
L
ρ
B
𝒜
.
Substituting (45) and (46) into (44), we conclude that there exists a constant K
>
0, independent of w, such that
(47)
∥
𝕄
s
♯
(
P
(
w
)
)
∥
L
B
𝒜
≤
K
∥
w
∥
L
ρ
B
𝒜
.
The proof of Corollary 17 has been completed.Virtually, we can obtain a global estimate about the composition operator by using Definition4.Corollary 18.
Suppose that the Young function𝒜 belongs to the G
(
p
,
q
,
C
)-class, and w
∈
C
∞
(
Λ
l
E
) is a solution to the nonhomogenous A-harmonic equation. Let the sharp maximal operator be noted by M
s
♯, P is the potential operator with its kernel function K
(
x
,
y
) satisfying standard estimates (1
≤
s
<
p, q
<
∞), and the bounded E
⊆
ℜ
m is the L
𝒜-averaging domains. Then, there exists a constant K
>
0 such that
(48)
∥
𝕄
s
♯
(
P
(
w
)
)
∥
L
E
𝒜
≤
K
∥
w
∥
L
E
𝒜
,
where B
0 is a fixed ball.



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*Source: 675464-2014-02-25.xml*