# Poincaré Inequalities for Composition Operators withL

Authors: Ru Fang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We establish the Poincaré-type inequalities for the composition of the homotopy operator, exterior derivative operator, and the projection operator withL
φ
-norm applied to the nonhomogeneous A-harmonic equation in L
φ
(
Ω
)-averaging domains.


---

## Body

## 1. Introduction

The purpose of the paper is to develop the Poincaré-type inequalities for the composition of the homotopy operatorT, exterior derivative operator d, and the projection operator H with L
φ-norm. These operators play critical roles in investigating the properties of the solutions to PDEs and in controlling oscillatory behavior of the solutions in domains [1–6]. We first establish the local Poincaré inequalities for the composition T
∘
d
∘
H in L
φ
(
Ω
)-averaging domains. Then, we prove the global Poincaré inequalities for the composition of T
∘
d
∘
H in L
φ
(
Ω
)-averaging domains.In this paper, we assumeΩ is a bounded and convex domain in ℝ
n
,
n
≥
2 and B
=
B
(
x
0
,
r
) is a ball that is centred at x
0 with r as its radius. For any σ
>
0, we use σ
B to denote the ball with centred at x
0 with radius σ
r. We do not distinguish the balls from the cubes in this paper. We use |
E
| to denote the Lebesgue measure of a set E
⊂
ℝ
n. We call ω a weight if ω
∈
L
loc
⁡
1
(
ℝ
n
) and ω
>
0 a.e. For a function u, we denote the average of u over B by u
B
=
(
1
/
|
B
|
)
∫
B
‍
u
d
x. Differential forms are extensions of functions in ℝ
n. For example, the function u
(
x
1
,
x
2
,
…
,
x
n
) is called a 0-form. Moreover, if u
(
x
1
,
x
2
,
…
,
x
n
) is differentiable, it is called a differential 0-form. The 1-form u
(
x
) in ℝ
n can be written as u
(
x
)
=
∑
i
=
1
n
‍
u
i
(
x
1
,
x
2
,
…
,
x
n
)
d
x
i. If the coefficient functions u
i
(
x
1
,
x
2
,
…
,
x
n
)
,
i
=
1,2
,
…
,
n, are differentiable, u
(
x
) is called a differential 1-form. Similarly, a differential k-form u
(
x
) is generated by {
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
k
}, k
=
1,2
,
…
,
n, that is, u
(
x
)
=
∑
I
‍
u
I
(
x
)
d
x
I = ∑
‍
u
i
1
i
2
⋯
i
k
(
x
)
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
k, where I
=
(
i
1
,
i
2
,
…
,
i
k
), 1
≤
i
1
<
i
2
<
⋯
<
i
k
≤
n. Let ∧
l
=
∧
l
(
ℝ
n
) be the set of all l-forms in ℝ
n, D
′
(
Ω
,
∧
l
) be the space of all differential l-forms on Ω and L
p
(
Ω
,
∧
l
) be the l-forms u
(
x
)
=
∑
I
‍
u
I
(
x
)
d
x
I on Ω satisfying ∫
Ω
‍
|
u
I
|
p
<
∞ for all ordered l-tuples I, l
=
1,2
,
…
,
n. We denote the exterior derivative by d
:
D
′
(
Ω
,
∧
l
)
→
D
′
(
Ω
,
∧
l
+
1
) for l
=
0,1
,
…
,
n
-
1, and define the Hodge star operator *
:
∧
k
→
∧
n
-
k as follows: if u
=
u
i
1
i
2
⋯
i
k
(
x
1
,
x
2
,
…
,
x
n
)
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
k
=
u
I
d
x
I, i
1
<
i
2
<
⋯
<
i
k, is a differential k-form, then *
u
=
*
(
u
i
1
i
2
⋯
i
k
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
k
)
=
(
-
1
)
∑
‍
(
I
)
u
I
d
x
J, where I
=
(
i
1
,
i
2
,
…
,
i
k
)
,  J
=
{
1,2
,
…
,
n
}
-
I, and ∑
‍
(
I
)
=
(
k
(
k
+
1
)
/
2
)
+
∑
i
=
1
k
‍
i
j. The Hodge codifferential operator d
*
:
D
′
(
Ω
,
∧
l
+
1
)
→
D
′
(
Ω
,
∧
l
) is given by d
*
=
(
-
1
)
n
l
+
1
*
d
* on D
′
(
Ω
,
∧
l
+
1
)
,
l
=
0,1
,
…
,
n
-
1.We useM to denote a bounded and convex domain on ℝ
n. Let ∧
l
M be the lth exterior power of the cotangent bundle, let C
∞
(
∧
l
M
) be the space of smooth l-forms on M, and 𝒲
(
∧
l
M
)
=
{
u
∈
L
loc
⁡
1
(
∧
l
M
)
:
u
has  generalized  gradient
}. The harmonic l-fields are defined by ℋ
(
∧
l
M
)
=
{
u
∈
𝒲
(
∧
l
M
)
:
d
u
=
d
*
u
=
0
,
u
∈
L
p
for  some
1
<
p
<
∞
}. The orthogonal complement of ℋ in L
1 is defined by ℋ
⊥
=
{
u
∈
L
1
:
〈
u
,
h
〉
=
0
for  all
h
∈
ℋ
}. Then, the Green's operator G is defined as G
:
C
∞
(
∧
l
M
)
→
ℋ
⊥
∩
C
∞
(
∧
l
M
) by assigning G
(
u
) as the unique element of ℋ
⊥
∩
C
∞
(
∧
l
M
) satisfying Poisson's equation Δ
G
(
u
)
=
u
-
H
(
u
), where H is the harmonic projection operator that maps C
∞
(
∧
l
M
) onto ℋ so that H
(
u
) is the harmonic part of u. See [7, 8] for more properties of these operators. The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. See [9, 10].The operatorK
y with the case y
=
0 was first introduced by Cartan in [11]. Then, it was extended to the following version in [12]. To each y
∈
Ω there corresponds a linear operator K
y
:
C
∞
(
Ω
,
∧
l
)
→
C
∞
(
Ω
,
∧
l
-
1
) defined by (
K
y
u
)
(
x
;
ξ
1
,
…
,
ξ
l
-
1
)
=
∫
0
1
‍
t
l
-
1
u
(
t
x
+
y − t
y
;
x
-
y
,
ξ
1
,
…
,
ξ
l
-
1
)
d
t and the decomposition u
=
d
(
K
y
u
)
+
K
y
(
d
u
). A homotopy operator T
:
C
∞
(
Ω
,
∧
l
)
→
C
∞
(
Ω
,
∧
l
-
1
) is defined by averaging K
y over all points y
∈
Ω
:
T
u
=
∫
Ω
‍
ϕ
(
y
)
K
y
(
d
u
), where ϕ
∈
C
0
∞
(
Ω
) is normalized so that ∫
‍
ϕ
(
y
)
d
y
=
1.We are particularly interested in a class of differential forms satisfying the well-known nonhomogeneousA-harmonic equation
(1)
d
*
A
(
x
,
d
u
)
=
B
(
x
,
d
u
)
,
where A
:
Ω
×
∧
l
(
ℝ
n
)
→
∧
l
(
ℝ
n
) and B
:
Ω
×
∧
l
(
ℝ
n
)
→
∧
l
-
1
(
ℝ
n
) 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
∈
Ω and all ξ
∈
∧
l
(
ℝ
n
). Here a
>
0 and b
>
0 are constants and 1
<
p
<
∞ is a fixed exponent associated with (1). A solution to (1) is an element of the Sobolev space W
loc
⁡
1
,
p
(
Ω
,
∧
l
-
1
) such that ∫
Ω
‍
A
(
x
,
d
u
)
·
d
φ
+
B
(
x
,
d
u
)
·
φ
=
0 for all φ
∈
W
loc
⁡
1
,
p
(
Ω
,
∧
l
-
1
) with compact support. If u is a function (0
-form) in ℝ
n, (1) reduces to
(3)
div
⁡
A
(
x
,
∇
u
)
=
B
(
x
,
∇
u
)
.
If the operator B
=
0, (1) becomes
(4)
d
*
A
(
x
,
d
u
)
=
0
,
which is called the homogeneous A-harmonic equation. Let A
:
Ω
×
∧
l
(
ℝ
n
)
→
∧
l
(
ℝ
n
) be defined by A
(
x
,
ξ
)
=
ξ
|
ξ
|
p
-
2 with p
>
1. Then, A satisfies the required conditions and d
*
A
(
x
,
d
u
)
=
0 becomes the p-harmonic equation d
*
(
d
u
|
d
u
|
p
-
2
)
=
0 for differential forms. Some results have been obtained in recent years about different versions of the A-harmonic equation; see [1, 2, 8, 9, 13–15].


## 2.  Main Results and Proofs

Definition 1.
Letφ be a continuously increasing convex function on [
0
,
∞
) with φ
(
0
)
=
0, and let Ω be a domain with μ
(
Ω
)
<
∞. If u is a measurable function in Ω, then we define the Orlicz norm of u by
(5)
∥
u
∥
φ
,
Ω
=
inf
⁡
{
k
>
0
:
∫
Ω
φ
(
|
u
(
x
)
|
k
)
d
x
≤
1
}
.
A continuously increasing function φ
:
[
0
,
∞
) with φ
(
0
)
=
0 is called an Orlicz function, and a convex Orlicz function φ is often called a Young function.From Definition1, it is easy to see that for any domain Ω
⊂
ℝ
n
(6)
∫
Ω
φ
(
|
u
(
x
)
|
∥
u
∥
φ
,
Ω
)
d
x
≤
1
if ∥
u
∥
φ
,
Ω is finite.Definition 2.
Letφ be an increasing convex function on [
0
,
∞
) with φ
(
0
)
=
0. We call a proper subdomain Ω
⊂
ℝ
n an Orlicz space L
φ
(
Ω
), if μ
(
Ω
)
<
∞ and there exists a constant C such that
(7)
∥
u
-
u
B
0
∥
φ
,
Ω
≤
C
sup
⁡
B
⊂
Ω
∥
u
-
u
B
∥
φ
,
B
for some ball B
0
⊂
Ω and all integrable functions u in Ω, where the supremum is over all balls B with B
⊂
Ω.Definition 3 (see [15]).
We say that a Young functionφ lies in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, if (
i
)
1
/
C
≤
φ
(
t
1
/
p
)
/
g
(
t
)
≤
C and (
ii
)
1
/
C
≤
φ
(
t
1
/
p
)
/
h
(
t
)
≤
C for all t
>
0, where g is a convex increasing function and h is a concave increasing function on [
0
,
∞
).From [15], we know that the class G
(
p
,
q
,
C
) contains some very interesting functions, such as φ
(
t
)
=
t
p and φ
(
t
)
=
t
p
log
⁡
+
α
(
t
)
,
p
≥
1
,
α
∈
ℝ, and each of φ
,
g and h is doubling in the sense that its values at t and 2
t are uniformly comparable for all t
>
0, and the consequent fact that
(8)
C
1
t
q
≤
h
-
1
(
φ
(
t
)
)
≤
C
2
t
q
,
C
1
t
p
≤
g
-
1
(
φ
(
t
)
)
≤
C
2
t
p
,
where C
1 and C
2 are constants. We will need the following reverse Ho
¨lder inequality.Lemma 4 (see [4]).
Letu be a solution of the nonhomogeneous A-harmonic equation (1) in a bounded and convex domain Ω and 0
<
s
,
t
<
∞. Then, there exists a constant C, independent of u, such that
(9)
∥
u
∥
s
,
B
≤
C
|
B
|
(
t
-
s
)
/
s
t
∥
u
∥
t
,
σ
B
for all balls B with σ
B
⊂
Ω for some σ
>
1.Lemma 5 (see [1]).
Letu be a solution of the nonhomogeneous A-harmonic equation (1) in a bounded and convex domain Ω. Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(10)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
s
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
s
,
B
for all balls B with B
⊂
Ω.Lemma 6 (see [1]).
Letu be a solution of the nonhomogeneous A-harmonic equation (1) in a bounded and convex domain Ω. Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(11)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
s
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
u
∥
s
,
σ
B
for all balls B with σ
B
⊂
Ω, where σ
>
1 is a constant.Theorem 7.
Letφ be a Young function in the class G
(
p
,
q
,
C
), 1
≤
p
<
q
<
∞, C
≥
1, and let Ω be a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the nonhomogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(12)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
σ
B
for some σ
>
1 and all balls B with σ
B
⊂
Ω.Proof.
For any constantk
>
0, from Lemmas 4 and 5, (i) in Definition 3, using the fact that φ is an increasing function, Jensen's inequality, and noticing that φ and g are doubling, we have
(13)
φ
(
1
k
(
∫
B
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
q
d
x
)
1
/
q
)
≤
φ
(
1
k
C
1
|
B
|
(
p
-
q
)
/
p
q
(
∫
σ
B
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
p
d
x
)
1
/
p
)
≤
φ
(
1
k
C
2
|
B
|
1
+
(
p
-
q
)
/
p
q
diam
⁡
(
B
)
(
∫
σ
B
|
d
u
|
p
d
x
)
1
/
p
)
≤
φ
(
(
1
k
p
C
2
p
|
B
|
p
+
(
p
-
q
)
/
q
(
diam
⁡
(
B
)
)
p
∫
σ
B
|
d
u
|
p
d
x
)
1
/
p
)
≤
C
3
g
(
1
k
p
C
2
p
|
B
|
p
+
(
p
-
q
)
/
q
(
diam
⁡
(
B
)
)
p
∫
σ
B
|
d
u
|
p
d
x
)
=
C
3
g
(
∫
σ
B
1
k
p
C
2
p
|
B
|
p
+
(
p
-
q
)
/
q
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
d
x
)
≤
C
3
∫
σ
B
‍
g
(
1
k
p
C
2
p
|
B
|
p
+
(
p
-
q
)
/
q
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
)
d
x
.
Since p
≥
1, then,1
+
(
p
-
q
)
/
p
q
>
0. Hence, we have |
B
|
1
+
(
p
-
q
)
/
p
q
≤
|
Ω
|
1
+
(
p
-
q
)
/
p
q
≤
C
4. From (i) in Definition 3, we find that g
(
t
)
≤
C
5
φ
(
t
1
/
p
). Thus,
(14)
∫
σ
B
g
(
1
k
p
C
2
p
|
B
|
p
+
(
p
-
q
)
/
q
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
)
d
x
≤
C
5
∫
σ
B
φ
(
1
k
C
2
|
B
|
1
+
(
p
-
q
)
/
p
q
diam
⁡
(
B
)
|
d
u
|
)
d
x
≤
C
5
∫
σ
B
φ
(
1
k
C
2
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
Combining (13) and (14) yields
(15)
φ
(
1
k
(
∫
B
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
q
d
x
)
1
/
q
)
≤
C
6
∫
σ
B
φ
(
1
k
C
2
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
Using Jensen's inequality for h
-
1, (8), and noticing that φ and h are doubling, we obtain
(16)
∫
B
‍
φ
(
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
k
)
d
x
=
h
(
h
-
1
(
∫
B
φ
(
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
(
k
)
-
1
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
d
x
∫
b
)
)
≤
h
(
∫
B
h
-
1
(
φ
(
(
k
)
-
1
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
)
d
x
∫
b
)
≤
h
(
C
7
∫
B
(
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
(
k
)
-
1
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
q
d
x
∫
b
)
≤
C
8
φ
(
(
C
7
∫
B
(
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
q
d
x
∫
b
)
1
/
q
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)

×

(
k
)
-
1
)
q
d
x
∫
b

)
1
/
q

)
≤
C
8
φ
(
1
k
(
C
7
∫
B
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
q
d
x
∫
b
)
1
/
q
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
q
d
x
∫
b
)
1
/
q
)
≤
C
9
φ
(
1
k
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
q
d
x
∫
s
)
1
/
q
(
∫
B
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
q
d
x
∫
s
)
1
/
q
)
.
Substituting (15) into (16) and noticing that φ is doubling, we have
(17)
∫
B
φ
(
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
k
)
d
x
≤
C
10
∫
σ
B
φ
(
1
k
C
2
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
≤
C
11
∫
σ
B
‍
φ
(
1
k
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
From Definition 2 and (17), we have the following version of Poincaré inequality with the Orlicz norm:
(18)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
σ
B
.
We have completed the proof of Theorem 7.Theorem 8.
Letφ be a Young function in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, and let Ω be a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the non-homogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(19)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
B
for some σ
>
1 and all balls B with σ
B
⊂
Ω.Proof.
For any constantk
>
0, from Lemma 5, (i) in Definition 3, using the fact that φ is an increasing function, Jensen's inequality, and noticing that φ and g are doubling, we have
(20)
φ
(
1
k
(
∫
B
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
p
d
x
)
1
/
p
)
≤
φ
(
1
k
C
1
|
B
|
diam
⁡
(
B
)
(
∫
B
|
d
u
|
p
d
x
)
1
/
p
)
≤
φ
(
(
1
k
p
C
1
p
|
B
|
p
(
diam
⁡
(
B
)
)
p
∫
B
|
d
u
|
p
d
x
)
1
/
p
)
≤
C
2
g
(
1
k
p
C
1
p
|
B
|
p
(
diam
⁡
(
B
)
)
p
∫
B
|
d
u
|
p
d
x
)
=
C
2
g
(
∫
B
1
k
p
C
1
p
|
B
|
p
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
d
x
)
≤
C
2
∫
B
g
(
1
k
p
C
1
p
|
B
|
p
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
)
d
x
.
Since p
≥
1, then |
B
|
≤
|
Ω
|
≤
C
3. From (i) in Definition 3, we find that g
(
t
)
≤
C
4
φ
(
t
1
/
p
). Thus,
(21)
∫
B
g
(
1
k
p
C
1
p
|
B
|
p
(
diam
⁡
(
B
)
)
p
|
d
u
|
p
)
d
x
≤
C
4
∫
B
φ
(
1
k
C
1
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
Combining (20) and (21) yields
(22)
φ
(
1
k
(
∫
B
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
p
d
x
)
1
/
p
)
≤
C
5
∫
B
φ
(
1
k
C
1
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
Using Jensen's inequality for g
-
1, (8), and noticing that φ and h are doubling, we obtain
(23)
∫
B
φ
(
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
k
)
d
x
=
g
(
g
-
1
(
∫
B
φ
(
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
(
k
)
-
1
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
d
x
∫
d
)
)
≤
g
(
∫
B
g
-
1
(
φ
(
(
k
)
-
1
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
)
d
x
∫
s
)
≤
g
(
C
6
∫
B
(
(
k
)
-
1
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
p
d
x
∫
s
)
≤
C
7
φ
(
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
×
(
k
)
-
1
)
p
d
x
∫
s
)
1
/
p

(

C
6
∫
B
(
(
k
)
-
1
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)

×

(
k
)
-
1
)
p
d
x

∫
s
)
1
/
p

)
≤
C
7
φ
(
1
k
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
p
d
x
∫
s
)
1
/
p
(
C
6
∫
B
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
p
d
x
∫
s
)
1
/
p
)
≤
C
8
φ
(
1
k
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
p
d
x
∫
s
)
1
/
p
(
∫
B
(
|
T
(
d
(
H
(
u
)
)
)
(
T
(
d
(
H
(
u
)
)
)
)
B
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
)
p
d
x
∫
s
)
1
/
p
)
.
Substituting (22) into (23) and noticing that φ is doubling, we have
(24)
∫
B
φ
(
|
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
|
k
)
d
x
≤
C
9
∫
B
φ
(
1
k
C
1
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
≤
C
10
∫
B
φ
(
1
k
|
B
|
diam
⁡
(
B
)
|
d
u
|
)
d
x
.
From Definition 2 and (24), we have the following version of Poincaré inequality with the Orlicz norm:
(25)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
B
.
We have completed the proof of Theorem 8.Using a similar method to the proof of Theorem8, we can establish the following version of Poincaré inequality with the Orlicz norm.Theorem 9.
Letφ be a Young function in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, and let Ω be a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the non-homogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(26)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
u
∥
φ
,
σ
B
for some σ
>
1 and all balls B with σ
B
⊂
Ω.Theorem 10.
Letφ be a Young function in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, and let Ω be a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the non-homogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that(27)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
0
∥
φ
,
Ω
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
Ω
,
where B
0
⊂
Ω is some fixed ball.Proof.
From definition of theL
φ
(
Ω
) and (12), we have
(28)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
0
∥
φ
,
Ω
≤
C
1
sup
⁡
B
⊂
Ω
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
1
sup
⁡
B
⊂
Ω
(
C
2
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
σ
B
)
≤
C
1
sup
⁡
B
⊂
Ω
(
C
2
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
Ω
)
≤
C
3
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
Ω
.
We have completed the proof of Theorem 10.Using a similar method to the proof of Theorem8, we obtain Theorem 11.Theorem 11.
Letφ be a Young function in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, and let Ω be a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the non-homogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(29)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
0
∥
φ
,
Ω
≤
C
|
B
|
diam
⁡
(
B
)
∥
u
∥
φ
,
Ω
,
where B
0
⊂
Ω is some fixed ball.It has been proved in [5] that any John domain is special L
φ
(
Ω
)-averaging domain. Hence, we have the following results.Corollary 12.
Letφ be a Young function in the class G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1, and let Ω be a bounded John domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the non-homogeneous A-harmonic (3) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(30)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
0
∥
φ
,
Ω
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
Ω
,
where B
0
⊂
Ω is some fixed ball.For some special convex function, we have the following theorems.Theorem 13.
Letφ
=
t
por  φ
=
t
p
log
⁡
α
(
e
+
t
)
∈
G
(
p
,
q
,
C
)
,
1
≤
p
<
q
<
∞
,
C
≥
1
,
α
∈
R a Young function, and Ω a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the nonhomogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(31)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
∥
φ
,
B
≤
C
|
B
|
diam
⁡
(
B
)
∥
d
u
∥
φ
,
σ
B
for some σ
>
1 and all balls B with σ
B
⊂
Ω.Theorem 14.
Letφ
=
t
p or φ
=
t
p
log
⁡
α
(
e
+
t
)
∈
G
(
p
,
q
,
C
),1
≤
p
<
q
<
∞
,
C
≥
1,α
∈
R a Young function, and Ω a bounded convex domain. Assume that φ
(
|
u
|
)
∈
L
loc
⁡
1
(
Ω
) and u is a solution of the nonhomogeneous A-harmonic (1) in Ω
,
φ
(
|
d
u
|
)
∈
L
loc
⁡
1
(
Ω
). Let H be the projection operator, and let T
:
C
∞
(
Ω
,
Λ
l
)
→
C
∞
(
Ω
,
Λ
l
-
1
) be the homotopy operator. Then, there exists a constant C, independent of u, such that
(32)
∥
T
(
d
(
H
(
u
)
)
)
-
(
T
(
d
(
H
(
u
)
)
)
)
B
0
∥
φ
,
Ω
≤
C
|
B
|
diam
⁡
(
B
)
∥
u
∥
φ
,
Ω
,
where B
0
⊂
Ω is some fixed ball.



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*Source: 818201-2014-04-01.xml*