Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 851236, 15 pages
doi:10.1155/2009/851236
Research Article
Weighted Norm Inequalities for Solutions to the
Nonhomogeneous A-Harmonic Equation
Haiyu Wen
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Haiyu Wen, [email protected]
Received 10 March 2009; Accepted 18 May 2009
Recommended by Shusen Ding
We first prove the local and global two-weight norm inequalities for solutions to the nonhomoge-
neous A-harmonic equation Ax, g duh d
v for differential forms. Then, we obtain some
weighed Lipschitz norm and BMO norm inequalities for differential forms satisfying the different
nonhomogeneous A-harmonic equations.
Copyright q 2009 Haiyu Wen. This is an open access article distributed 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
In the recent years, the A-harmonic equations for differential forms have been widely
investigated, see 1, and many interesting and important results have been found, such as
some weighted integral inequalities for solutions to the A-harmonic equations; see 2–7.
Those results are important for studying the theory of differential forms and both qualitative
and quantitative properties of the solutions to the different versions of A-harmonic equation.
In the different versions of A-harmonic equation, the nonhomogeneous A-harmonic equation
Ax, g duh d
v has received increasing attentions, in 8 Ding has presented
2 Journal of Inequalities and Applications
equations. These results can be used to study the basic properties of the solutions to the
nonhomogeneous A-harmonic equations.
Now, we first introduce related concepts and notations.
Throughout this paper we assume that Ω is a bounded connected open subset of R
n
.
We assume that B is a ball in Ω with diameter diamB and σB is the ball with the same
center as B with diamσBσ diamB.Weuse|E| to denote the Lebesgue measure of
E. We denote w a weight if w ∈ L
1
loc
R
n
and w>0 a.e Also in general dμ wdx.
For 0 <p<∞, we write f ∈ L
p
E, w
α
if the weighted L
p
-norm of f over E satisfies
f
p,E,w
α
E
|fx|
p
i
2
···i
l
xdx
i
1
∧dx
i
2
∧···∧dx
i
l
with w
I
∈ L
p
Ω, R for all ordered l-tuples I i
1
,i
2
, ,i
l
,
1 ≤ i
1
<i
2
< ··· <i
l
Ω, Λ
l
→ D
Ω, Λ
l1
for l 0, 1, ,n−1. Its formal adjoint operator d
: D
Ω, Λ
l1
→
D
Ω, Λ
l
is given by d
−1
nl1
don D
Ω, Λ
l1
, l 0, 1, 2, ,n−1. A differential l-form
u ∈ D
Ω, Λ
In this paper we consider solutions to the nonhomogeneous A-harmonic equation
A
x, g du
h d
v 1.1
for differential forms, where g,h ∈ D
Ω, Λ
l
and A : Ω × Λ
l
R
n
→ Λ
l
R
n
satisfies the
following conditions:
|
A
x, ξ
|
≤ a
|
Definition 1.1. We call u and v a pair of conjugate A-harmonic tensor in Ω if u and v satisfy
the conjugate A-harmonic equation
A
x, du
d
v 1.3
in Ω,andA
−1
exists in Ω, we call u and v conjugate A-harmonic tensors in Ω.
We also consider solutions to the equation of t he form
d
A
x, dw
B
x, dw
, 1.4
Journal of Inequalities and Applications 3
here A : Ω × Λ
l
R
n
→ Λ
≥
|
ξ
|
p
,
|
B
x, ξ
|
≤ b
|
ξ
|
p−1
, 1.5
for almost every x ∈ Ω and all ξ ∈ Λ
l
R
n
.Herea, b > 0 are constants and 1 <p<∞ is a
fixed exponent associated with 1.4. A solution to 1.4 is an element of the Sobolev space
W
1
p,loc
Ω, Λ
l−1
In this section, we will extend Lemma 2.3,seein8, to new version with A
r,λ
Ω weight both
locally and globally.
Definition 2.1. We say a pair of weights w
1
x,w
2
x satisfies the A
r,λ
Ω-condition in a
domain Ω and write w
1
x,w
2
x ∈ A
r,λ
Ω for some λ ≥ 1and1<r<∞ with 1/r 1/r
1, if
sup
B⊂Ω
1
|
B
|
B
See 9 for properties of A
r,λ
Ω-weights. We will need the following generalized
H
¨
older’s inequality.
Lemma 2.2. Let 0 <α<∞, 0 <β<∞, and s
−1
α
−1
β
−1
,iff and g are measurable functions on
R
n
,then
fg
s,Ω
≤
f
α,Ω
·
v ∈ L
q
B, Λ
l
. Moreover, there exist constants C
1
and C
2
, independent of u and v, such that
d
v
q
q,B
≤ C
1
h
q
q,B
g
p,B
d
v
q
q,B
,
2.3
for all balls B with B ⊂ Ω ⊂ R
n
.
Theorem 2.4. Let u and v be a pair of solutions to the nonhomogeneous A-harmonic equation 1.1
in a domain Ω ⊂ R
n
. Assume that w
1
x,w
2
x ∈ A
r,λ
Ω for some λ ≥ 1 and 1 <r<∞ with
1/r 1/r
1. Then, there exists a constants C, independent of u and v, such that
d
t,B,w
αt/s
2
|
du
|
p/q
t,B,w
αt/s
2
, 2.4
for all balls B with B ⊂ Ω ⊂ R
n
.Hereα is any positive constant with λ>αr, s qλ − α/λ, and
t sλ/λ − αrqsλ/sλ − qαr − 1. Note that 2.4 can be written as the following symmetric
form:
|
B
|
−1/s
t,B,w
αt/s
2
|
du
|
p/q
t,B,w
αt/s
2
. 2.5
Proof. Choose s qλ − α/λ < q,since1/s 1/q q − s/qs,usingH
¨
older inequality, we
find that
d
v
α/s
1
s
dx
1/s
≤
B
|
d
v
|
q
dx
1/q
B
w
α/s
1
qs/q−s
dx
T−1
N
i1
|t
i
|
T
and Lemma 2.3,weobtain
d
v
q,B
≤ C
1
h
q,B
g
p/q
p,B
q
dx
1/q
≤
B
|
h
|
t
w
αt/s
2
dx
1/t
B
1
w
2
αqt/st−q
dx
p,B
B
g
p
w
αt/ks
2
w
−αt/ks
2
dx
1/p
≤
B
g
k
1
w
2
λ/r−1
dx
k−p/kp
.
2.9
We know that
k − p
kp
αt
r − 1
sλ
·
sλ
sλp αpt
r − 1
α
r − 1
p/q
k,B,w
αt/s
2
·
B
1
w
2
λ/r−1
dx
αr−1/sλ
. 2.11
6 Journal of Inequalities and Applications
Note that
g
p/q
k,B,w
αt/s
2
psλ/pqsλαpqtr−1
B
g
psλαtr−1/sλ
w
αt/s
2
dx
sλ/qsλαqtr−1
.
2.12
Since
r − 1
αt sλ
sλt
q
, 2.13
then,
g
p/q
t,B,w
αt/s
2
.
2.14
Combining 2.11 and 2.14,weobtain
g
p/q
p,B
≤
g
p/q
|
du
|
p/q
t,B,w
αt/s
2
·
B
1
w
2
λ/r−1
dx
αr−1/sλ
. 2.16
Combining 2.6 and 2.7 gives
d
v
1
dx
α/sλ
. 2.17
Substituting 2.8, 2.15,and2.16 into 2.17, we have
d
v
s,B,w
α
1
≤ C
1
h
t,B,w
αt/s
2
g
1
dx
α/sλ
B
1
w
2
λ/r−1
dx
αr−1/sλ
.
2.18
Journal of Inequalities and Applications 7
Since w
1
,w
2
∈ A
r,λ
Ω, then
B
w
B
1
w
2
λ/r−1
dx
r−1
⎞
⎠
α/sλ
⎛
⎜
⎝
|
B
|
1/λr
1
|
B
|
B
w
λ
⎞
⎟
⎠
αr/s
≤ C
2
|
B
|
αr/sλ
.
2.19
Putting 2.19 into 2.18, we obtain the desired result
d
v
s,B,w
α
1
≤ C
3
|
B
|
αr/sλ
h
The proof of Theorem 2.4 has been completed.
Using the same method, we have the following two-weighted L
s
-estimate for du.
Theorem 2.5. Let u and v be a pair of solutions to the nonhomogeneous A-harmonic equation 1.1
in a domain Ω ⊂ R
n
. Assume that w
1
x,w
2
x ∈ A
r,λ
Ω for some λ ≥ 1 and 1 <r<∞ with
1/r 1/r
1. Then, there exists a constants C, independent of u and v, such that
du
s,B,w
α
1
≤ C
|
B
|
αr/sλ
q/p
t,B,w
αt/s
2
, 2.21
for all balls B with B ⊂ Ω ⊂ R
n
.Hereα is any positive constant with λ>αr, s pλ − α/λ, and
t sλ/λ − αrpsλ/sλ − pαr − 1.
It is easy to see that the inequality 2.21 is equivalent to
|
B
|
−1/s
du
s,B,w
α
1
≤ C
|
B
|
−1/t
|
q/p
t,B,w
αt/s
2
. 2.22
As applications of the local results, we prove the following global norm comparison
theorem.
Lemma 2.6. Each Ω has a modified Whitney cover of cubes V {Q
i
} such that
i
Q
i
Ω,
Q∈V
χ
√
5/4Q
≤ Nχ
Ω
,
2.23
8 Journal of Inequalities and Applications
1
and C
2
, independent of u and v, such that
d
v
s,Ω,w
α
1
≤ C
1
h
t,Ω,w
αt/s
2
g
1
≤ C
2
g
t,Ω,w
αt/s
2
|
h
|
q/p
t,Ω,w
αt/s
2
v
|
s
w
α
1
dx
1/s
≤
B∈V
B
|
d
v
|
s
w
α
1
dx
1/s
≤
t,B,w
αt/s
2
g
p/q
t,B,w
αt/s
2
|
du
|
p/q
p/q
t,Ω,w
αt/s
2
|
du
|
p/q
t,Ω,w
αt/s
2
χ
√
5/4B
≤ C
2
t,Ω,w
αt/s
2
B∈V
χ
√
5/4B
≤ C
3
h
t,Ω,w
αt/s
2
g
Ω-condition in a domain Ω write
wx ∈ A
r
Ω for some 1 <r<∞ with 1/r 1/r
1, if
sup
B⊂Ω
1
|
B
|
B
wdx
1/r
⎛
⎝
1
|
B
|
B
1
w
1
loc
Ω, Λ
l
, l 0, 1, 2, ,n. We write ω ∈ locLip
k
Ω, Λ
l
,0≤ k ≤ 1, if
ω
locLip
k
,Ω
sup
σB⊂Ω
|
B
|
−nk/n
ω − ω
B
1,B
< ∞, 3.1
for some σ ≥ 1.
Similarly, we write ω ∈ BMOΩ, Λ
l
,w
α
,0≤
k ≤ 1, if
ω
locLip
k
,Ω,w
α
sup
σB⊂Ω
μ
B
−nk/n
ω − ω
B
1,B,w
α
< ∞. 3.3
Similarly, for ω ∈ L
1
loc
for some σ>1, where Ω is a bounded domain, the measure μ is defined by dμ wx
α
dx, w
is a weight, and α is a real number.
10 Journal of Inequalities and Applications
We need the following classical Poincar
´
e inequality; see 10.
Lemma 3.3. Let u ∈ D
Ω, Λ
l
and du ∈ L
q
B, Λ
l1
,thenu − u
B
is in W
1
q
B, Λ
l
with 1 <q<∞
and
u − u
B
q,B
q,σB
, 3.6
for all balls B with σB ⊂ Ω.
We need the following local weighted Poincar
´
e inequality for A-harmonic tensors.
Theorem 3.5. Let u ∈ D
Ω, Λ
l
be an A-harmonic tensor in a domain Ω ⊂ R
n
and du ∈
L
s
Ω, Λ
l1
, l 0, 1, 2, ,n. Assume that σ>1, 1 <s<∞, and w
1
x,w
2
x ∈ A
r,λ
Ω
for some λ ≥ 1 and 1 <r<∞ with 1/r 1/r
1. Then, there exists a constant C, independent of u,
such that
u − u
B
|
u − u
B
|
s
w
α
1
dx
1/s
B
|
u − u
B
|
w
α/s
1
s
dx
t,B
B
w
λ
1
dx
α/λs
.
3.8
Taking m λs/λ αr −1, then m<s<t, using Lemmas 3.4 and 3.3 and the same method
as 2, Proof of Theorem 2.12,weobtain
u − u
B
s,B,w
α
1
≤ C
2
|
B
|
11/n
|
B
αm/s
2
w
−αm/s
2
dx
1/m
σB
|
du
|
w
α/s
2
w
−α/s
2
m
dx
1/m
≤
s,B,w
α
1
≤ C
2
|
B
|
11/nm−t/mt
du
s,σB,w
α
2
w
1
α/s
λ,B
1
w
2
λ/r−1,σB
≤
⎛
⎝
σB
w
λ
1
dx
σB
1
w
2
λ/r−1
dx
r−1
⎞
⎠
α/λs
⎛
⎜
λr
/r
dx
⎞
⎠
1/λr
⎞
⎟
⎠
rα/s
≤ C
3
|
B
|
rα/λs
.
3.12
Combining 3.11 and 3.12 gives
u − u
B
s,B,w
α
1
≤ C
0. 3.14
Finally, we obtain the desired result
u − u
B
s,B,w
α
1
≤ C
4
|
B
|
11/n
du
s,σB,w
α
2
. 3.15
This ends the proof of Theorem 3.5.
12 Journal of Inequalities and Applications
Similarly, if setting w
1
xw
2
x and λ 1inTheorem 3.5, we obtain Theorem 2.12
in 2. And we choose w
l
be an A-harmonic tensor in a domain Ω ⊂ R
n
, and all c ∈
D
Ω, Λ
l
with dc 0, and du ∈ L
s
Ω, Λ
l1
, l 0, 1, 2, ,n − 1. Assume that 1 <s<∞
and w
1
x,w
2
x ∈ A
r,λ
Ω for some λ ≥ 1 and 1 <r<∞ with w
1
x ≥ >0 for any x ∈ Ω.
Then, there exist constants C and C
, independent of u, such that
u − c
locLip
k
B
w
α
1
dx ≥
B
α
dx C
1
|B| implies that
1
μ
1
B
≤
C
2
|
B
|
, 3.19
for any ball B.Using3.7 and the H
¨
older inequality with 1 1/s s − 1/s, we have
B
1
s/s−1
dμ
1
s−1/s
μ
1
B
s−1/s
u − u
B
s,B,w
α
1
≤
μ
1
B
μ
1
B
−nk/n
u − c −
u − c
B
1,B,w
α
1
sup
σB⊂Ω
μ
1
B
−1−k/n
≤ C
4
sup
σB⊂Ω
|
B
|
−1/s−k/n11/n
du
s,σB,w
α
2
≤ C
4
sup
σB⊂Ω
|
Ω
|
−1/s−k/n11/n
du
s,σB,w
,Ω,w
α
1
sup
σB⊂Ω
μ
1
B
−1
u − c −
u − c
B
1,B,w
α
1
≤ sup
σB⊂Ω
μ
1
μ
1
B
−nk/n
u − u
B
1,B,w
α
1
.
3.22
From 3.21 we find
u − c
,Ω,w
α
1
≤ C
1
u − c
Theorem 3.8. Let u ∈ D
Ω, Λ
l
be an A-harmonic tensor in a domain Ω ⊂ R
n
, and all c ∈
D
Ω, Λ
l
with dc 0, and du ∈ L
s
Ω, Λ
l1
, l 0, 1, 2, ,n − 1. Assume that 1 <s<∞
14 Journal of Inequalities and Applications
and wx ∈ A
r
Ω for r>1 with wx ≥ >0 for any x ∈ Ω. Then, there exist constants C and C
,
independent of u, such that
u − c
locLip
k
,Ω,w
α
du
s,Ω
,
u − c
,Ω
≤ C
du
s,Ω
.
3.27
Using Lemma 3.6, we can also obtain the following theorem.
Theorem 3.9. Let u and v be a pair of conjugate A-harmonic tensor in a domain Ω ⊂ R
n
,then
du ∈ L
p
Ω, Λ
l
,μ if and only if d
v ∈ L
q
Ω, Λ
l
u − c
,Ω,w
α
≤ C
d
v
q/p
qt/p,Ω,w
αt/s
,
3.28
where k and α are positive constants with 0 ≤ k ≤ 1 and αr < 1,fors 1 − αp, t s/1 − αr
ps/s − αpr − 1.
Proof. From 3.25, we have
u − c
locLip
k
,Ω,w
α
≤ C
1
α
≤ C
3
du
s,Ω,w
α
≤ C
4
d
v
q/p
qt/p,Ω,w
αt/s
. 3.31
Journal of Inequalities and Applications 15
If w ≡ 1, we have
u − c
locLip
k
,Ω
≤ C
d
Illinois Journal of Mathematics, vol. 46, no. 1, pp. 199–205, 2002.
3 B. Liu, “A
r
λ-weighted Caccioppoli-type and Poincar
´
e-type inequalities for A-harmonic tensors,”
International Journal of Mathematics and Mathematical Sciences, vol. 31, no. 2, pp. 115–122, 2002.
4 Y. Xing, “Weighted Poincar
´
e-type estimates for conjugate A-harmonic tensors,” Journal of Inequalities
and Applications, no. 1, pp. 1–6, 2005.
5 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.
6 S. Ding and Y. Ling, “Weighted norm inequalities for conjugate A-harmonic tensors,” Journal of
Mathematical Analysis and Applications, vol. 203, no. 1, pp. 278–288, 1996.
7 S. Ding and P. Shi, “Weighted Poincar
´
e-type inequalities for differential forms in L
s
μ-averaging
domains,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 200–215, 1998.
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 C. J. Neugebauer, “Inserting A
p
-weights,” Proceedings of the American Mathematical Society, vol. 87, no.
4, pp. 644–648, 1983.
10 S. Ding, “Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic
equations on Riemannian manifolds,” Proceedings of the American Mathematical Society, vol. 132, no.
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