Volume 2009, Article ID 851236, 15 pagesdoi:10.1155/2009/851236 Research Article Weighted Norm Inequalities for Solutions to the Haiyu Wen Department of Mathematics, Harbin Institute of
Trang 1Volume 2009, Article ID 851236, 15 pages
doi:10.1155/2009/851236
Research Article
Weighted Norm Inequalities for Solutions to the
Haiyu Wen
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Haiyu Wen,wenhy@hit.edu.cn
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 du h 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.
Copyrightq 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, see1, 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
A x, g du h d v has received increasing attentions, in 8 Ding has presented some estimates to such equation In this paper, we extend some estimates that Ding has presented in8 into the two-weight case Our results are more general, so they can be used broadly
It is well-known that the Lipschitz norm supQ⊂Ω |Q| −1−k/n u − u Q1,Q, where the
supremum is over all local cubes Q, as k → 0 is the BMO norm supQ⊂Ω |Q|−1u − u Q1,Q,
so the natural limit of the space locLipkΩ as k → 0 is the space BMOΩ InSection 3,
we establish a relation between these two norms and L p-norm We first present the local
two-weight Poincar´e inequality for A-harmonic tensors Then, as the application of this inequality
and the result in8, we prove some weighted Lipschitz norm inequalities and BMO norm inequalities for differential forms satisfying the different nonhomogeneous A-harmonic
Trang 2equations 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 thatΩ is a bounded connected open subset of Rn
We assume that B is a ball in Ω with diameter diamB and σB is the ball with the same center as B with diamσB σ diamB We use |E| to denote the Lebesgue measure of
E We denote w a weight if w ∈ L1
locRn 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 w x α dx1/p
< ∞, where α is a real number A differential l-form
ω on Ω is a schwartz distribution on Ω with value in ΛlRn, we denote the space of differential l-forms by D
Ω, Λ l We write L p Ω, Λ l for the l-forms wx I w I xdx I
w i1i2···i l xdx i1∧ dx i2∧ · · · ∧ dx i l with w I ∈ L p Ω, R for all ordered l-tuples I i1, i2, , i l,
1 ≤ i1 < i2 < · · · < i l ≤ n, l 0, 1, , n Thus L p Ω, Λ l is a Banach space with norm
w p,Ω Ω|wx| p dx1/p ΩI |w I x|2p/2 dx1/p We denote the exterior derivative by
d : DΩ, Λ l → D
Ω, Λ l1 for l 0, 1, , n−1 Its formal adjoint operator d : DΩ, Λ l1 →
DΩ, Λ l is given by d −1nl1 d on DΩ, Λ l1 , l 0, 1, 2, , n − 1 A differential l-form
u ∈ DΩ, Λ l is called a closed form if du 0 in Ω Similarly, a differential l 1-form
v ∈ D
Ω, Λ l1 is called a coclosed form if d v 0 The l-form ω B ∈ D
B, Λ l is defined by
ω B |B|−1B ω ydy, l 0 and ω B dTω, l 1, 2, , n, for all ω ∈ L p B, Λ l , 1 ≤ p < ∞, here T is a homotopy operator, for its definition, see8
Then, we introduce some A-harmonic equations.
In this paper we consider solutions to the nonhomogeneous A-harmonic equation
A
for differential forms, where g, h ∈ D
Ω, Λ l and A : Ω × Λ lRn → ΛlRn satisfies the following conditions:
for almost every x ∈ Ω and all ξ ∈ Λ lRn Here a > 0 is a constant and 1 < p < ∞ is a fixed
exponent associated with1.1 and p−1 q−1 1 Note that if we choose g h 0 in 1.1, then1.1 will reduce to the conjugate A-harmonic equation Ax, du d v.
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
inΩ, and A−1exists inΩ, we call u and v conjugate A-harmonic tensors in Ω.
We also consider solutions to the equation of the form
Trang 3here A :Ω × ΛlRn → ΛlRn and B : Ω × Λ lRn → Λl−1Rn satisfy the conditions:
for almost every x ∈ Ω and all ξ ∈ Λ lRn Here a, b > 0 are constants and 1 < p < ∞ is a
fixed exponent associated with1.4 A solution to 1.4 is an element of the Sobolev space
W p,loc1 Ω, Λ l−1 such that
Ω
for all ϕ ∈ W1
p,loc Ω, Λ l−1 , with compact support.
Definition 1.2 We call u an A-harmonic tensor in Ω if u satisfies the A-harmonic equation
1.4 in Ω
In this section, we will extendLemma 2.3, see in8, to new version with A r,λΩ weight both locally and globally
Definition 2.1 We say a pair of weights w1x, w2x satisfies the A r,λΩ-condition in a domainΩ and write w1x, w2x ∈ A r,λ Ω for some λ ≥ 1 and 1 < r < ∞ with 1/r 1/r
1, if
sup
B⊂Ω
1
|B|
B
w1λ dx
1/λr⎛
⎝ 1
|B|
B
1
w2
λr/r
dx
⎞
⎠
1/λr
for any ball B⊂ Ω
See 9 for properties of A r,λΩ-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, if f and g are measurable functions on
Rn , then
fg
s,Ω≤f
α,Ω·g
for anyΩ ∈ Rn
Trang 4We also need the following lemma; see8.
Lemma 2.3 Let u and v be a pair of solutions to the nonhomogeneous A-harmonic equation 1.1
in a domain Ω ⊂ Rn If g ∈ L p B, Λ l and h ∈ L q B, Λ l , then du ∈ L p B, Λ l if and only if
d v ∈ L q B, Λ l Moreover, there exist constants C1and C2, independent of u and v, such that
d vq q,B ≤ C1
h q q,Bgp
p,B du p
p,B
,
du p p,B ≤ C2
h q q,Bgp
p,B d vq
q,B
,
2.3
for all balls B with B⊂ Ω ⊂ Rn
Theorem 2.4 Let u and v be a pair of solutions to the nonhomogeneous A-harmonic equation 1.1
in a domainΩ ⊂ Rn Assume that w1x, w2x ∈ 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 vs,B,w α ≤ C|B| αr/sλ
h t,B,w αt/s
2 gp/q
t,B,w αt/s
2
|du| p/q
t,B,w αt/s
2
for all balls B with B⊂ Ω ⊂ Rn Here α is any positive constant with λ > αr, s qλ − α/λ, and
t sλ/λ − αr qsλ/sλ − qαr − 1 Note that 2.4 can be written as the following symmetric
form:
|B| −1/s d vs,B,w α ≤ C|B| −1/t
h t,B,w αt/s
2 gp/q
t,B,w αt/s
2
|du| p/q
t,B,w αt/s
2
Proof Choose s qλ − α/λ < q, since 1/s 1/q q − s/qs, using H¨older inequality, we
find that
d vs,B,w α
B
|d v|s w α
1xdx 1/s
B
|d v |w α/s
1
s
dx
1/s
≤
B
|d v|q dx
1/q
B
w α/s1 qs/q−s
dx
q−s/qs
≤ d vq,B
B
w λ1dx
α/λs
.
2.6
Trang 5Applying the elementary inequality|N
i1 t i|T ≤ N T−1N
d vq,B ≤ C1
h q,Bgp/q
p,B du p/q
p,B
Choose t qsλ/sλ − qαr − 1 > q, using H¨older inequality with 1/q 1/t t − q/qt again
yields
h q,B
B
|h|w α/s
2 w −α/s2 q
dx
1/q
≤
B
|h| t w2αt/s dx
1/t
B
1
w2
αqt/st−q
dx
t−q/qt
h t,B,w αt/s
2
B
1
w2
λ/r−1
dx
αr−1/λs
.
2.8
Then, choosing k p αptr − 1/sλ > p, using H¨older inequality once again, we have
g
p,B
B
gp
w2αt/ks w −αt/ks2 dx
1/p
≤
B
gk
w αt/s2 dx
1/k
B
1
w2
αtp/sk−q
dx
k−q/kp
g
k,B,w αt/s2
B
1
w2
λ/r−1
dx
k−p/kp
.
2.9
We know that
k − p
kp αt r − 1
sλp αptr − 1
α r − 1
sp · st
sλ αtr − 1
α r − 1q
spλ ,
2.10
and hence
gp/q p,B ≤gp/q
k,B,w αt/s
2
·
B
1
w2
λ/r−1
dx
αr−1/sλ
Trang 6Note that
gp/q k,B,w αt/s2
B
gk
w2αt/s dx
p/kq
B
gpsλαptr−1/sλ w αt/s2 dx
psλ/pqsλαpqtr−1
B
gpsλαtr−1/sλ w2αt/s dx
sλ/qsλαqtr−1
.
2.12
Since
r − 1αt sλ sλt
then,
gp/q k,B,w αt/s
2
B
gpt/q
w2αt/s dx
1/t
gp/q
t,B,w αt/s2 .
2.14
Combining2.11 and 2.14, we obtain
gp/q p,B ≤gp/q
t,B,w αt/s
2
·
B
1
w2
λ/r−1
dx
αr−1/sλ
Using the similar method, we can easily get that
du p/q p,B ≤|du| p/q
t,B,w αt/s
2
·
B
1
w2
λ/r−1
dx
αr−1/sλ
Combining2.6 and 2.7 gives
d vs,B,w α ≤ C1
h q,Bgp/q
p,B du p/q
p,B
B
w λ1dx
α/sλ
Substituting2.8, 2.15, and 2.16 into 2.17, we have
d vs,B,w α ≤ C1
h t,B,w αt/s
2 gp/q
t,B,w αt/s
2
|du| p/q
t,B,w αt/s
2
·
B
w λ1dx
α/sλ
B
1
w2
λ/r−1
dx
αr−1/sλ
.
2.18
Trang 7Sincew1, w2 ∈ A r,λΩ, then
B
w1λ dx
α/sλ
B
1
w2
λ/r−1
dx
αr−1/sλ
⎛
⎝
B
w λ1dx
B
1
w2
λ/r−1
dx
r−1⎞
⎠
α/sλ
⎛
⎜
⎝|B| 1/λr
1
|B|
B
w λ
1dx
1/λr
|B| 1/λr
⎛
⎝ 1
|B|
B
1
w2
λr/r
dx
⎞
⎠
1/λr⎞
⎟
αr/s
≤ C2|B| αr/sλ
2.19
Putting2.19 into 2.18, we obtain the desired result
d vs,B,w α ≤ C3|B| αr/sλ
h t,B,w αt/s
2 |g| p/q
t,B,w αt/s
2
|du| p/q
t,B,w αt/s
2
The proof ofTheorem 2.4has 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 domainΩ ⊂ Rn Assume that w1x, w2x ∈ 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 ≤ C|B| αr/sλ g
t,B,w αt/s2 |h| q/p
t,B,w2αt/s|d v|q/p
t,B,w αt/s2 , 2.21
for all balls B with B⊂ Ω ⊂ Rn Here α is any positive constant with λ > αr, s pλ − α/λ, and
t sλ/λ − αr psλ/sλ − pαr − 1.
It is easy to see that the inequality2.21 is equivalent to
|B| −1/s du s,B,w α ≤ C|B| −1/t g
t,B,w2αt/s|h| q/p
t,B,w αt/s2 |d v|q/p
t,B,w αt/s2 . 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
Trang 8for all x∈ Rn and some N > 1 and if Q i ∩ Q j / ∅, then there exists a cube R (this cube does not need
be a member of V) in Q i ∩ Q j such that Q i ∩ Q j ⊂ NR
Theorem 2.7 Let u and v be a pair of solutions to the nonhomogeneous A-harmonic equation 1.1
in a bounded domainΩ ⊂ Rn Assume that w1x, w2x ∈ A r,λ Ω for some λ ≥ 1 and 1 < r < ∞
with 1/r 1/r 1 Then, there exist constants C1and C2, independent of u and v, such that
d vs,Ω,w α ≤ C1
h t,Ω,w αt/s
2 gp/q
t,Ω,w αt/s
2
|du| p/q
t,Ω,w αt/s
2
Here α is any positive constant with λ > αr, s qλ−α/λ, t sλ/λ−αr qsλ/sλ−qαr −1,
and
du s,Ω,w α ≤ C2
g
t,Ω,w αt/s
2 |h| q/p
t,Ω,w αt/s
2
|d v|q/p
t,Ω,w αt/s
2
for s pλ − α/λ and t sλ/λ − αr psλ/sλ − pαr − 1.
Proof ApplyingTheorem 2.4andLemma 2.6, we have
d vs,Ω,w α
Ω|d v|s w1α dx
1/s
B∈V
B
|d v|s w α1dx
1/s
B∈V
B
|d v|s w α1dx
1/s
χ√ 5/4B
≤ C1
B∈V
|B| αr/sλ
h t,B,w αt/s
2 gp/q
t,B,w αt/s2 |du| p/q
t,B,w αt/s2 χ√ 5/4B
≤ C1
B∈V
|Ω|αr/sλ
h t,Ω,w αt/s
2 gp/q
t,Ω,w αt/s
2
|du| p/q
t,Ω,w αt/s
2
χ√ 5/4B
≤ C2
h t,Ω,w αt/s
2 gp/q
t,Ω,w αt/s
2
|du| p/q
t,Ω,w αt/s
2 B∈V
χ√ 5/4B
≤ C3
h t,Ω,w αt/s
2 gp/q
t,Ω,w αt/s
2
|du| p/q
t,Ω,w αt/s
2
.
2.26
Since Ω is bounded The proof of inequality 2.24 has been completed Similarly, using
proof ofTheorem 2.7
Trang 9Definition 2.8 We say the weight w x satisfies the A rΩ-condition in a domain Ω write
w x ∈ A r Ω for some 1 < r < ∞ with 1/r 1/r
1, if
sup
B⊂Ω
1
|B|
B
w dx
1/r⎛
⎝ 1
|B|
B
1
w
r/r
dx
⎞
⎠
1/r
for any ball B⊂ Ω
We see that A r,λ Ω-weight reduce to the usual A r Ω-weight if w1x w2x and
λ 1; see 10
And, if w1x w2x and λ 1 inTheorem 2.7, it is easy to obtain Theorems 4.2 and 4.4 in8
3 Estimates for Lipschitz Norms and BMO Norms
In11 Ding has presented some estimates for the Lipchitz norms and BMO norms In this section, we will prove another estimates for the Lipchitz norms and BMO norms
Definition 3.1 Let ω ∈ L1
locΩ, Λ l , l 0, 1, 2, , n We write ω ∈ locLip k Ω, Λ l , 0 ≤ k ≤ 1, if
ωlocLip
k ,Ω sup
σB⊂Ω |B| −nk/n ω − ω B1,B < ∞, 3.1
for some σ≥ 1
Similarly, we write ω ∈ BMOΩ, Λ l if
ω ,Ω sup
for some σ ≥ 1 When ω is a o-form, 3.2 reduces to the classical definition of BMOΩ
We also discuss the weighted Lipschitz and BMO norms
Definition 3.2 Let ω ∈ L1
locΩ, Λ l , w α , l 0, 1, 2, , n We write ω ∈ locLip k Ω, Λ l , w α, 0 ≤
k≤ 1, if
ωlocLipk ,Ω,w α sup
σB⊂Ω
μ B−nk/n ω − ω B1,B,w α < ∞. 3.3
Similarly, for ω ∈ L1
locΩ, Λ l , w α , l 0, 1, 2, , n We write ω ∈ BMOΩ, Λ l , w α, if
ω ,Ω,w α sup
σB⊂Ω
μ B−1ω − ω B1,B,w α < ∞, 3.4
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.
Trang 10We need the following classical Poincar´e inequality; see10.
Lemma 3.3 Let u ∈ DΩ, Λ l and du ∈ L q B, Λ l1 , then u − u B is in W1B, Λ l with 1 < q < ∞
and
u − u Bq,B ≤ Cn, q
We also need the following lemma; see2
Lemma 3.4 Suppose that u is a solution to 1.4, σ > 1 and q > 0 There exists a constant C,
depending only on σ, n, p, a, b, and q, such that
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 ∈ DΩ, 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 w1x, w2x ∈ A r,λΩ
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 Bs,B,w α ≤ C|B||B| 1/n du s,σB,w α , 3.7
for all balls B with σB ⊂ Ω Here α is any constant with 0 < α < λ.
Proof Choose t λs/λ − α, since 1/s 1/t t − s/st, using H¨older inequality, we find that
u − u Bs,B,w α
B
|u − u B|s w1α dx
1/s
B
|u − u B |w α/s
1
s
dx
1/s
≤
B
|u − u B|t dx
1/t
B
w α/s1 st/t−s
dx
t−s/st
u − u Bt,B
B
w1λ dx
α/λs
.
3.8
Taking m λs/λ αr − 1, then m < s < t, using Lemmas3.4and3.3and the same method
as2, Proof of Theorem 2.12, we obtain
u − u Bs,B,w α ≤ C2|B|11/n|B| m−t/mt du m,σB w1α/s