Báo cáo hóa học: " Research Article ¨ The Reverse Holder Inequality for the Solution to p-Harmonic Type System" potx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (520.7 KB, 15 trang )
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 397340, 15 pages
doi:10.1155/2008/397340
Research Article
The Reverse H
¨
older Inequality for
the Solution to p-Harmonic Type System
Zhenhua Cao,
1
Gejun Bao,
1
Ronglu Li,
1
and Haijing Zhu
2
1
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2
College of Mathematics and Physics, Shan Dong Institute of Light Industry, Jinan 250353, China
Correspondence should be addressed to Gejun Bao,
Received 6 July 2008; Revised 9 September 2008; Accepted 5 November 2008
Recommended by Shusen Ding
Some inequalities to A-harmonic equation Ax, dud
∗
v have been proved. The A-harmonic
equation is a particular form of p-harmonic type system Ax, a dub d
∗
v only when a 0
and b 0. In this paper, we will prove the Poincar
´
e inequality and the reverse H
¨
older inequality
for the solution to the p-harmonic type system.
Copyright q 2008 Zhenhua Cao et al. This is an open access article d istributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Recently, amount of work about the A-harmonic equation for the differential forms has been
done. In fact, the A-harmonic equation is an important generalization of the p-harmonic
equation in R
n
, p>1, and the p-harmonic equation is a natural extension of the usual Laplace
equation see 1 for the details. The reverse H
¨
older inequalities have been widely studied
and frequently used in analysis and related fields, including partial differential equations and
the theory of elasticity see 2. In 1999, Nolder gave the reverse H
¨
older inequality for the
solution to the A-harmonic equation in 3,anddifferent versions of Caccioppoli estimates
have been established in 4–6. In 2004, D’Onofrio and Iwaniec introduced the p-harmonic
type system in 7, which is an important extension of the conjugate A-harmonic equation. In
2007, Ding proved the following inequality in 8.
Theorem A. Let u, v be a pair of solutions to Ax, g duh d
∗
v in a domain Ω ⊂ R
n
.If
g ∈ L
p
B, Λ
L
and h ∈ L
q
B, Λ
L
,thendu ∈ L
p
B, Λ
L
if and only if d
∗
v ∈ L
q
B, Λ
L
. Moreover,
there exist constants C
1
,C
2
independent of u and v, such that
d
∗
v
q
q,B
≤ C
1
h
q
q,B
g
p
p,B
du
p
p,B
,
du
p
p,B
≤ C
2
h
q
q,B
g
p
p,B
d
∗
v
q
q,B
∀B ⊂ σB ⊂ Ω.
1.1
2 Journal of Inequalities and Applications
In this paper, we will prove the Poincar
´
e inequality see Theorem 2.5 and the reverse
H
¨
older inequality for the solution to the p-harmonic type system see Theorem 3.5.Nowlet
us see some notions and definitions about the p-harmonic type system.
Let e
1
,e
2
, ,e
n
denote the standard orthogonal basis of R
n
. For l 0, 1, ,n, we
denote by Λ
l
Λ
l
R
n
the linear space of all l-vectors, spanned by the exterior product e
I
e
i
1
∧e
i
2
∧···∧e
i
l
corresponding to all ordered l-tuples I i
1
,i
2
, ,i
l
,1≤ i
1
<i
2
< ···<i
l
≤ n.
The Grassmann algebra Λ⊕Λ
l
is a graded algebra with respect to the exterior products. For
α
α
I
e
I
∈ Λ and β
β
I
e
I
∈ Λ, then its inner product is obtained by
α, β
α
I
β
I
, 1.2
with the summation over all I i
1
,i
2
, ,i
l
and all integers l 0, 1, ,n. The Hodge star
operator ∗: Λ → Λ is defined by the rule
∗1 e
i
1
∧ e
i
2
∧···∧e
i
n
,
α ∧∗β β ∧∗α α, β∗1 ∀α, β ∈ Λ.
1.3
Hence, the norm of α ∈ Λ can be given by
|α|
2
α, α ∗α ∧∗α ∈ Λ
0
R. 1.4
Throughout this paper, Ω ⊂ R
n
is an open subset, for any constant σ>1, Q denotes
a cube such that Q ⊂ σQ ⊂ Ω, where σQ denotes the cube whose center is as same as Q
and diamσQσ diam Q.Wesayα
α
I
e
I
∈ Λ is a differential l-form on Ω, if every
coefficient α
I
of α is Schwartz distribution on Ω. The space spanned by differential l-form
on Ω is denoted by D
Ω, Λ
l
. We write L
p
Ω, Λ
l
for the l-form α
α
I
dx
I
on Ω with
α
I
∈ L
p
Ω for all ordered l-tuple I.ThusL
p
Ω, Λ
l
is a Banach space with the norm
α
p,Ω
Ω
|α|
p
dx
1/p
Ω
I
|α
I
|
2
p/2
dx
1/p
. 1.5
Similarly W
k,p
Ω, Λ
l
denotes those l-forms on Ω with all coefficients in W
k,p
Ω. We denote
the exterior derivative by
d : D
Ω, Λ
l
−→ D
Ω, Λ
l1
, for l 0, 1, 2, ,n, 1.6
and its formal adjoint operator the Hodge codi fferential operator
d
∗
: D
Ω, Λ
l
−→ D
Ω, Λ
l−1
. 1.7
The operators d and d
∗
are given by the formulas
dα
I
dα
I
∧ dx
I
,d
∗
−1
nl1
∗d∗. 1.8
Zhenhua Cao et al. 3
The following two definitions appear in 7.
Definition 1.1. The Hodge system holds:
Ax, a dub d
∗
v, 1.9
where a ∈ L
p
Ω, Λ
l
and b ∈ L
q
Ω, Λ
l
,isap-harmonic type system if A is a mapping from
Ω × Λ
l
to Λ
l
satisfying
1 x → Ax, ξ is measurable in x ∈ Ω for every ξ ∈ Λ
l
;
2 ξ → Ax, ξ is continuous in ξ ∈ Λ
l
for almost every x ∈ Ω;
3 Ax, tξt
p−1
Ax, ξ for every t ≥ 0;
4 KAx, ξ − Ax, ζ,ξ− ζ≥|ξ − ζ|
2
|ξ| |ζ|
p−2
;
5 |Ax, ξ − Ax, ζ|≤K|ξ − ζ||ξ| |ζ|
p−2
for almost every x ∈ Ω and all ξ, ζ ∈ Λ
l
, where K ≥ 1 is a constant. It should be noted that
Ax, ∗ : Ω × Λ
l
→ Λ
l
is invertible and its inverse denoted by A
−1
satisfies similar conditions
as A but with H
¨
older conjugate exponent q in place of p.
Definition 1.2. If 1.9 is a p-harmonic type system, then we say the equation
d
∗
Ax, a dud
∗
b 1.10
is a p-harmonic type equation.
The following definition appears in 9.
Definition 1.3. Adifferential form u is a weak solution for 1.10 in Ω if u satisfies
Ω
Ax, a du,dϕ
d
∗
b, ϕ
≡ 0 1.11
for every ϕ ∈ W
k,p
Ω, Λ
l−1
with compact support.
We can find that if we let a 0andb 0, then the p-harmonic type system
Ax, a dub d
∗
v 1.12
becomes
Ax, dud
∗
v. 1.13
It is the conjugate A-harmonic equation, where the mapping A : Ω × Λ
l
→ Λ
l
satisfies the
following conditions:
Ax, ξ
≤ a|ξ|
p−1
,
Ax, ξ,ξ
≥|ξ|
p
. 1.14
4 Journal of Inequalities and Applications
If we let Ax, ξ|ξ|
p−2
ξ, then the conjugate A-harmonic equation becomes the form
|du|
p−2
du d
∗
v. 1.15
It is the conjugate p-harmonic equation.
So we can see that the conjugate p-harmonic equation and the conjugate A-harmonic
equation are the specific p-harmonic type system.
Remark 1.4. It should be noted that the mapping Ax, ∗ in p-harmonic system Ax, a du
b d
∗
v, is invertible. If we denote its inverse by A
−1
x, ∗, then the mapping A
−1
x, ∗ : Λ
l
→
Λ
l
satisfies similar conditions as A but with H
¨
older conjugate exponent q in place of p.
2. The Poincar
´
e inequality
In this section, we will introduce the Poincar
´
e inequality for the differential forms.
Now first let us see a lemma, which can be found in 9,Section4 for the details.
Lemma 2.1. Let D be a bounded, convex domain in R
n
.Toeachy ∈ D there corresponds a linear
operator K
y
: C
∞
D, Λ
l
→ C
∞
D, Λ
l−1
defined by
K
y
ω
x; ξ
1
, ,ξ
l−1
1
0
t
l−1
ω
tx y − ty; x − y, ξ
1
, ,ξ
l−1
dt, 2.1
and the decomposition
ω d
K
y
ω
K
y
dω2.2
holds at any point y ∈ D.
We construct a homotopy operator T : C
∞
D, Λ
l
→ C
∞
D, Λ
l−1
by averaging K
y
over all
points y ∈ D:
Tω
D
ϕyK
y
ωdy, 2.3
where ϕ form C
∞
D is normalized so that
ϕydy 1. It is obvious that ω dK
y
ωK
y
dω
remains valid for the operator T :
ω dTωTdω. 2.4
We define the l-forms ω
D
∈ D
D, Λ
l
by ω
D
|D|
−1
D
ωydy for l 0 and ω
D
dTω for
l 1, 2, ,n,and all ω ∈ W
1,p
D, Λ
l
, 1 <p<∞.
The following definition can be found in [9, page 34].
Definition 2.2. For ω ∈ D
D, Λ
l
, the vector valued differential form
∇ω
∂ω
∂x
1
, ,
∂ω
∂x
n
2.5
Zhenhua Cao et al. 5
consists of differential forms ∂ω/∂x
i
∈ D
D, Λ
l
, where the partial differentiation is applied
to coefficients of ω.
The proof of 9, Proposition 4.1 implies the following inequality.
Lemma 2.3. For any ω ∈ L
p
D, Λ
l
, it holds that
∇Tω
p,D
≤ Cn, pω
p,D
2.6
for any ball or cube D ∈ R
n
.
The following Poincar
´
e inequality can be found in [2].
Lemma 2.4. If u ∈ W
1,p
0
Ω, then there is a constant C Cn, p > 0 such that
1
|B|
B
|u|
pχ
dx
1/pχ
≤ Cr
1
|B|
B
|∇u|
p
dx
1/p
, 2.7
whenever B Bx
0
,r is a ball in R
n
,wheren ≥ 2 and χ 2 for p ≥ n, χ np/n − p for p<n.
Theorem 2.5. Let u ∈ D
D, Λ
l
, and du ∈ L
p
D, Λ
l1
. Then, u − u
D
is in L
χp
D, Λ
l
and
1
|D|
D
|u − u
D
|
pχ
dx
1/pχ
≤ Cn, p, ldiamD
1
|D|
D
|du|
p
dx
1/p
2.8
for any ball or cube D ∈ R
n
,whereχ 2 for p ≥ n and χ np/n − p for 1 <p<n.
Proof. We know Tduu − u
D
. Now we suppose u − u
Q
Tdu
I
u
I
dx
I
, where I
i
1
, ,i
l1
take over all l 1-tuples. So we have
∇Tdu
∂u
∂x
1
, ,
∂u
∂x
n
I
∂u
I
∂x
1
dx
I
, ,
I
∂u
I
∂x
n
dx
I
. 2.9
So we have
1
|D|
D
u − u
D
pχ
dx
1/pχ
1
|D|
D
I
u
I
dx
I
pχ
dx
1/pχ
1
|D|
D
I
u
I
2
pχ/2
dx
1/pχ
.
2.10
6 Journal of Inequalities and Applications
By the inequality
n
i1
a
i
2
1/2
≤
n
i1
a
i
≤ n
1/2
n
i1
a
i
2
1/2
2.11
for any a
i
≥ 0, and the Minkowski inequality, we have
1
|D|
D
I
u
I
2
pχ/2
dx
1/pχ
≤
I
1
|D|
D
u
I
pχ
dx
1/pχ
. 2.12
According to the Poincar
´
e inequality, we have
I
1
|D|
D
|u
I
|
pχ
dx
1/pχ
≤ C
1
n, pdiamD
I
1
|D|
D
|∇u
I
|
p
dx
1/p
. 2.13
Combining 2.10, 2.12,and2.13, we can obtain
1
|D|
D
u − u
D
pχ
dx
1/pχ
≤ C
1
n, pdiamD
I
1
|D|
D
|∇u
I
|
p
dx
1/p
. 2.14
By 2.9 we have
∇Tdu
p,D
∂u
∂x
1
, ,
∂u
∂x
n
p,D
D
∂u
∂x
1
, ,
∂u
∂x
n
p
dx
1/p
D
n
i1
∂u
∂x
i
2
p/2
dx
1/p
D
n
i1
I
∂u
I
∂x
i
2
p/2
dx
1/p
D
I
n
i1
∂u
I
∂x
i
2
p/2
dx
1/p
.
2.15
Zhenhua Cao et al. 7
Combining 2.11 and 2.15, then we have
∇Tdu
p,D
D
I
n
i1
∂u
I
∂x
i
2
p/2
dx
1/p
≥
C
l1
n
−1/2
D
I
n
i1
∂u
I
∂x
i
2
1/2
p
dx
v
1/p
≥
C
l1
n
−1/2
D
I
n
i1
∂u
I
∂x
i
2
p/2
dx
1/p
≥
C
l1
n
−1/2
C
l1
n
−p−1/p
I
D
n
i1
∂u
I
∂x
i
2
p/2
dx
1/p
≥
C
2
n, p, l
−1
I
D
∇u
I
p
dx
1/p
,
2.16
where C
2
n, p, lC
l1
n
1/2p−1/p
. Now combining 2.14, 2.16,and2.6, we can get
1
|D
D
u − u
D
|
pχ
dx
1/pχ
≤ C
1
n, pdiamD
I
1
|D|
D
∇u
I
p
dx
1/p
≤ C
1
n, p, lC
2
n, p, l
1
|D|
1/p
∇Tdu
p,D
≤ C
3
n, p, ldiamD
1
|D|
D
|du|
p
dx
1/p
.
2.17
3. The reverse H
¨
older inequality
In this section, we will prove the reverse H
¨
older inequality for the solution of the p-harmonic
type system. Before we prove the reverse H
¨
older inequality, let us first see some lemmas.
Lemma 3.1. If f, g ≥ 0 and for any nonnegative η ∈ C
∞
0
Ω, it holds
Ω
ηf dx ≤
Ω
gdx, 3.1
then for any h ≥ 0:
Ω
ηfh dx ≤
Ω
ghdx. 3.2
8 Journal of Inequalities and Applications
Proof. Let μ be a measure in X, f be a nonnegative μ-measurable function in a measure space
X, using the standard representation theorem, we have
X
f
q
dμ q
∞
0
t
q−1
μx : fx >tdt 3.3
for any 0 <t<q.Now, we let μE
E
ηf dx and νE
E
gdxthen, we can obtain
Ω
ηfh dx
∞
0
h>t
ηf dx dt ≤
∞
0
h>t
gdxdt
Ω
ghdx. 3.4
So Lemma 3.1 is proved.
Lemma 3.2. If u, v is a pair of solution to the p-harmonic type system 1.9, then it holds
Ω
|ηda|
p
dx ≤ C
Ω
|a dudη|
p
dx 3.5
for any nonnegative η ∈ C
∞
0
Ω and where C C
l1
n
p
.
Proof. Since u, v is a pair of solutions to Ax, a dub d
∗
v, it is also the solution to
A
−1
x, b d
∗
va du, where A
−1
x, ∗ is the inverse Ax, ∗. Now, we suppose that da
I
ω
I
dx
I
and let ϕ
1
−
I
η signω
I
dx
I
.Byusingϕ ϕ
1
and dϕ
1
I
signω
I
dη ∧ dx
I
in
1.11, we can obtain
Ω
A
−1
x, b d
∗
v
,dϕ
1
da, ϕ
1
dx ≡ 0. 3.6
That is,
Ω
da,
I
η sign
ω
I
dx
I
dx
Ω
A
−1
x, b d
∗
v
, −
I
sign
ω
I
dη ∧ dx
I
dx. 3.7
In other words,
Ω
I
η
ω
I
dx
Ω
A
−1
x, b d
∗
v
, −
I
sign
ω
I
dη ∧ dx
I
dx. 3.8
By the elementary inequality
n
i1
a
i
2
1/2
≤
n
i1
a
i
, 3.9
Zhenhua Cao et al. 9
we have
Ω
η|da|dx
Ω
η
I
ω
I
2
1/2
dx≤
Ω
I
η|ω
I
|dx
Ω
A
−1
x, b d
∗
v
, −
I
sign
ω
I
dη ∧ dx
I
dx.
3.10
Using the inequality
|a, b| ≤ |a||b|, 3.11
3.10 becomes
Ω
η|da|dx ≤
Ω
A
−1
x, b d
∗
v
I
sign
ω
I
dη ∧ dx
I
≤
Ω
A
−1
x, b d
∗
v
I
sign
ω
I
|dη|dx
C
l1
n
Ω
A
−1
x, b d
∗
v
|dη|dx
C
l1
n
Ω
|a du||dη|dx,
3.12
where I takes over all l 1-tuples for dη ∈ Λ
l1
,thusithasC
l1
n
numbers at most. Now we
let f |da| and g C
l1
n
|a du||dη|. In the subset {x : fη g}, we have
{x:fηg}
|ηda|
p
dx ≤
{x:fηg}
|a dudη|
p
dx. 3.13
In the subset {x : fη
/
g},leth |fη|
p
−|g|
p
/fη − g, then we easily obtain h>0. So by
Lemma 3.1, we have
{x:fη
/
g}
hfη dx ≤
{x:fη
/
g}
hg dx. 3.14
That is to say
{x:fη
/
g}
hfη − gdx ≤ 0, 3.15
10 Journal of Inequalities and Applications
that is,
{x:fη
/
g}
|fη|
p
dx ≤
fη
/
g
|g|
p
dx. 3.16
Combining 3.13 and 3.16, we have
Ω
|fη|
p
dx ≤
Ω
|g|
p
dx, 3.17
that is,
Ω
|ηda|
p
dx ≤
Ω
C
l1
n
a dudη
p
dx. 3.18
So Lemma 3.2 is proved.
The following lemma appears in 2.
Lemma 3.3. Suppose that 0 <q<p<s≤∞, ξ ∈ R, and that B Bx
0
,r is a ball. If a nonnegative
function v ∈ L
p
B, dμ satisfies
1
μ
λB
λB
v
s
dμ
1/s
≤ C1 − λ
ξ
1
μ
B
B
v
p
dμ
1/p
3.19
for each ball B
Bx
0
,r
with r
≤ r and for all 0 <λ<1,then
1
μλB
λB
v
s
dμ
1/s
≤ C1 − λ
ξ/θ
1
μB
B
v
q
dμ
1/q
∀0 <λ<1. 3.20
Here C>0 is a constant depending on p, q, s and θ ∈ 0, 1 is such that 1/p θ/q 1 − θ/s.
The following lemma appears in 10.
Lemma 3.4. Let u, v be a pair of solutions of the p-harmonic type system on domain Ω,thenwe
have a constant C only depending on K, n, p, and l, such that
ηdu
p,Ω
≤ C
u − cdη
p,Ω
ηa
p,Ω
, 3.21
Zhenhua Cao et al. 11
where c is any closed form (i.e., dc 0) and for any η ∈ C
∞
0
Ω. Also we have a constant C
only
depending on K, n, q, s uch that
ηd
∗
v
q,Ω
≤ C
v − c
dη
q,Ω
ηb
q,Ω
, 3.22
where c
is any coclosed form (i.e., d
∗
c
0) and q is the conjugate exponent of p.
Theorem 3.5. If u, v is a pair of solutions to the p-harmonic type system, then there exists a constant
C>0 dependent on K, p, n, and l, such that
1
|Q|
Q
u − u
Q
a
∞,Q
s
dx
1/s
≤ C
1 − σ
−1
−tχ/pχ−1
diam Q 1
χ/χ−1
×
1
|σQ|
σQ
u − u
σQ
a
∞,σQ
t
dx
1/t
3.23
for any 0 <s, t<∞, σ>1 and all cubes with Q ⊂ σQ ⊂ Ω,whereχ>1 is the Poincar
´
e constant.
Proof. Suppose that the center of Q is x
0
and diam Q r,0<λ σ
−1
< 1. Let
r
m
λ 1 − λ2
−m
,m 0, 1, 2, 3.24
Then r
m
is decreasing and λ<r
m
< 1. So we have u
Q
|
r
m
Q
u
r
m
Q
, for any m ∈ 0, 1, 2,
Let η
m
∈ C
∞
0
r
m
Q be a nonnegative function such that η
m
1inr
m1
Q,0 ≤ η
m
≤ 1in
r
m
Q − r
m1
Q. |dη
m
|≤1 − λ
−1
2
m
r
−1
.Givenanyt ≥ 0andletω
m
|u − u
Q
| a
∞,Q
1t/p
η
m
,
then we have
du
m
1
t
p
u − u
Q
a
∞,Q
t/p
η
m
d
u − u
Q
u − u
Q
a
∞,Q
1t/p
dη
m
.
3.25
By the Minkowski inequality, we can obtain
r
m
Q
du
m
p
dx
1/p
≤
r
m
Q
u − u
Q
a
∞,Q
pt
dη
m
p
dx
1/p
p t
p
r
m
Q
d
u − u
Q
p
u − u
Q
a
∞,Q
t
η
m
p
dx
1/p
.
3.26
We assume that u − u
Q
I
a
I
dx
I
, then we have |u − u
Q
|
I
a
2
I
1/2
.Ifu − u
Q
is zero, then
we have |d|u − u
Q
|| 0 |∇Tdu|.Ifu − u
Q
is not equal zero, and the proof of 2.15 implies
12 Journal of Inequalities and Applications
that |∇Tdu|
I
n
i1
|∂a
I
/∂x
i
|
2
1/2
d
u − u
Q
∇
u − u
Q
∂
u − u
Q
∂x
1
, ,
∂
u − u
Q
∂x
n
n
i1
∂
u − u
Q
∂x
i
2
1/2
n
i1
∂
u − u
Q
∂x
i
2
1/2
n
i1
∂
I
a
2
I
1/2
∂x
i
2
1/2
n
i1
1
I
a
2
I
I
a
I
∂a
I
∂x
i
2
1/2
≤
n
i1
1
I
a
2
I
I
a
2
I
I
∂a
I
∂x
i
2
1/2
n
i1
I
∂a
I
∂x
i
2
1/2
n
i1
I
∂a
I
∂x
i
2
1/2
∇Tdu
∇u − u
Q
.
3.27
So we have
d
u − u
Q
≤
∇Tdu
. 3.28
For any η ∈ C
∞
0
Ω, according to 2.6, we have
η∇Tdω
p,D
≤ Cn, p max
x∈D
ηdω
p,D
. 3.29
By the similar method as Lemma 3.1, we can prove the following inequality:
r
m
Q
d
u − u
Q
p
u − u
Q
a
∞,Q
t
η
m
p
dx
1/p
≤
r
m
Q
η
m
p
∇Tdu
p
u − u
Q
a
∞,Q
t
dx
1/p
≤ Cn, pmax
x∈D
η
p
m
r
m
Q
η
m
p
|du|
p
u − u
Q
a
∞,Q
t
dx
1/p
3.30
Zhenhua Cao et al. 13
for any η ∈ C
∞
0
Ω.ByLemma 3.1 and 3.21, we can obtain
r
m
Q
η
m
p
|du|
p
u − u
Q
a
∞,Q
t
dx
1/p
≤ 2C
r
m
Q
η
m
p
|a|
p
u − u
Q
a
∞,Q
t
dx
1/p
2C
r
m
Q
dη
m
p
u − u
Q
a
∞,Q
pt
dx
1/p
≤ 2C
r
m
Q
η
m
p
a
p
∞,Q
u − u
Q
a
∞,Q
t
dx
1/p
2C
r
m
Q
dη
m
p
u − u
Q
a
∞,Q
pt
dx
1/p
≤ 2C
r
m
Q
η
m
p
u − u
Q
a
∞,Q
pt
dx
1/p
2C
r
m
Q
dη
m
p
u − u
Q
a
∞,Q
pt
dx
1/p
.
3.31
Combining 3.26, 3.30,and3.31, by the values of η
m
, we have
r
m
Q
du
m
p
dx
1/p
≤ C
1
p t
1 1 − λ
−1
2
m
r
−1
r
m
Q
u − u
Q
a
∞,Q
pt
dx
1/p
.
3.32
For η
m
1inr
m1
Q and 0 ≤ η
m
≤ 1inr
m
Q − r
m1
Q, and as we have |r
m
|/r
m1
|λ 1 −
λ2
−m
|/λ 1 − λ2
−m−1
≤ 2, so we have |r
m
Q|/|r
m1
Q|≤2
n
. By the Poincar
´
e inequality, we
know
1
|r
m1
Q|
r
m1
Q
u − u
Q
a
∞,Q
χpt
dx
1/pχ
≤
1
r
m1
Q
r
m
Q
η
pχ
m
u − u
Q
a
∞,Q
χpt
dx
1/pχ
≤
1
r
m1
Q
r
m
Q
u
m
pχ
dx
1/pχ
≤ 2
n
1
r
m
Q
r
m
Q
u
m
pχ
dx
1/pχ
≤ C
2
r
m
r
1
r
m
Q
r
m
Q
du
m
p
dx
1/p
≤ C
3
r
m
rp t
1 1 − λ
−1
2
m
r
−1
r
m
Q
u − u
Q
a
∞,Q
pt
dx
1/p
≤ C
3
p t1 − λ
−1
2
m
1 r
r
m
Q
u − u
Q
a
∞,Q
pt
dx
1/p
.
3.33
14 Journal of Inequalities and Applications
Now we set κ p t, then by computation, we obtain
1
r
m1
Q
r
m1
Q
u − u
Q
a
∞,Q
κχ
dx
1/κχ
≤
C
3
p/κ
κ
p/κ
1 − λ
−p/κ
2
pm/κ
r 1
p/κ
×
1
r
m
Q
r
m
Q
u − u
Q
a
∞,Q
κ
dx
1/κ
.
3.34
Since this inequality holds for all κ>p, it can be applied with κ κ
m
pχ
m
. And we
can easily prove 1/|Q|
Q
|f|
p
dx
1/p
is increasing with p and its limit is ess sup
Q
|f|.Soby
iterating we arrive at the desired inequality for q p:
ess sup
λQ
u − u
Q
a
∞,Q
≤ lim
m →∞
1
r
m
Q
r
m
Q
u − u
Q
a
∞,Q
κ
m
χ
dx
1/κ
m
χ
≤ C
4
1 − λ
−1
1 r
Σ
∞
i0
χ
−m
∞
m0
2
mχ
−m
∞
m0
pχ
m
χ
−m
×
1
|Q|
Q
u − u
Q
a
∞,Q
p
dx
1/p
≤ C
5
1 − λ
−χ/χ−1
r 1
χ/χ−1
1
|Q|
Q
u − u
Q
a
∞,Q
p
dx
1/p
.
3.35
We can observe that the constants C
5
and χ are independent of x
0
and r in 3.35,thus
3.35 holds not only in the cube Q Qx
0
,r but also in each ball inside Q. By Lemma 3.5
we can obtain
1
|λQ|
λQ
u − u
Q
a
∞,Q
s
dx
1/s
≤ C
5
1 − λ
−θχ/χ−1
r 1
χ/χ−1
×
1
|Q|
Q
|u − u
Q
| a
∞,Q
t
dx
1/t
3.36
for any 0 <t<p<s≤∞, where θ ts − p/ps − t. So we have θ ≤ t/p for any
0 <t<p<s≤∞. Since 1/|Q|
Q
|f|
p
dx
1/p
is increasing with p,
1
|λQ|
λQ
u − u
Q
a
∞,Q
s
dx
1/s
≤ C
5
1 − λ
−tχ/pχ−1
r 1
χ/χ−1
×
1
|Q|
Q
|u − u
Q
| a
∞,σQ
t
dx
1/t
3.37
Zhenhua Cao et al. 15
for any 0 <s<∞ and 1 <p<t<∞. Combining 3.36 and 3.37, we have
1
|Q|
Q
u − u
Q
a
∞,Q
s
dx
1/s
≤ C
6
1 − λ
−tχ/pχ−1
r 1
χ/χ−1
×
1
|σQ|
σQ
|u − u
σQ
| a
∞,σQ
t
dx
1/t
3.38
for any 0 <s, t<∞ and σ>1 such that σQ ⊂ Ω. Theorem 3.5 is proved.
Acknowledgment
This work is supported by the NSF of China Grants nos. 10671046 and 10771044.
References
1 S. Ding, “Some examples of conjugate p-harmonic differential forms,” Journal of Mathematical Analysis
and Applications, vol. 227, no. 1, pp. 251–270, 1998.
2 J. Heinonen, T. Kilpel
¨
ainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,
Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1993.
3 C. A. Nolder, “Hardy-Littlewood theorems for A-harmonic tensors,” Illinois Journal of Mathematics,
vol. 43, no. 4, pp. 613–632, 1999.
4 G. Bao, “A
r
λ-weighted integral inequalities for A-harmonic tensors,” Journal of Mathematical
Analysis and Applications, vol. 247, no. 2, pp. 466–477, 2000.
5 S. Ding, “Weighted Caccioppoli-type estimates and weak reverse H
¨
older inequalities for A-harmonic
tensors,” Proceedings of the American Mathematical Society, vol. 127, no. 9, pp. 2657–2664, 1999.
6 X. Yuming, “Weighted integral inequalities for solutions of the A-harmonic equation,” Journal of
Mathematical Analysis and Applications, vol. 279, no. 1, pp. 350–363, 2003.
7 L. D’Onofrio and T. Iwaniec, “The p-harmonic transform beyond its natural domain of definition,”
Indiana University Mathematics Journal, vol. 53, no. 3, pp. 683–718, 2004.
8 S. Ding, “Local and global norm comparison theorems for solutions to the nonhomogeneous A-
harmonic equation,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1274–1293,
2007.
9 T. Iwaniec and A. Lutoborski, “Integral estimates for null Lagrangians,” Archive for Rational Mechanics
and Analysis, vol. 125, no. 1, pp. 25–79, 1993.
10 Z. Cao, G. Bao, and R. Li, “The Caccioppoli estimate for the s olution to the p-harmonic type system,”
to appear in the Proceedings of the 6th International Conference on Differential Equations and
Dynamical Systems.
