Weighted Decomposition Estimates for Differential Forms

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Volume 2010, Article ID 649340,17pages doi:10.1155/2010/649340

Research Article

Weighted Decomposition Estimates for

Differential Forms

Yong Wang and Guanfeng Li

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Yong Wang,[email protected]

Received 28 October 2009; Revised 5 March 2010; Accepted 24 March 2010

Academic Editor: Shusen Ding

Copyrightq2010 Y. Wang and G. Li. 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.

After introducing the definition ofAβ_r,λ-weights, we establish theArΩ-weighted decomposition

estimates andAβ_r,λΩ-weighted Caccioppoli-type estimates forA-harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domainΩ⊂ Rn_{. These results can}

be used to study the integrability of diﬀerential forms and to estimate the integrals for diﬀerential forms.

1. Introduction

Lete1, e2, . . . , endenote the standard orthogonal basis ofRn. Suppose thatΛl ΛlRnis the

linear space of alll-vectors, spanned by the exterior producteI ei1∧ei2∧· · ·∧eilcorresponding to all orderedl-tuplesI i1, i2, . . . , il, 1≤i1 < i2 < · · ·< il ≤ n. Throughout this paper, we

always assume thatΩis an open subset ofRn_{. We use} _D_M,_Λl_{to denote the space of all}

diﬀerentiall-forms andLp_M,_Λl_{to denote the}_l_-forms_ω_x

IωIdxI

ωi1i2···ilxdx1∧

dx2∧ · · · ∧dxlonMwith the coeﬃcientωI ∈LpΩ,Rfor all orderedl-tuplesI, whereMis

a manifold. ThusLp_M,_Λl_{is a Banach space with the norm}

ωp,M

M

|ωx|pdx

1/p

⎛ ⎝

M

I

|ωIx|2

p/2

dx

⎞ ⎠

1/p

. 1.1

For a diﬀerential l-form ω ∈ DΩ,∧l_{, its vector-valued di}_ﬀ_{erential form} _∇ω

∂ω/∂x1, ∂ω/∂x2, . . . , ∂ω/∂xn is composed of the diﬀerentiall-form ∂ω/∂xi ∈ DΩ,∧l,

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suppose thatLp₁Ω,∧l_{is the space consisting of all}_∂ω/∂x

i∈LpΩ,∧l, theith one of∇ω. We

denote the exterior diﬀerential operator ofl-forms byd:DΩ,∧l _→ _D_Ω_,_∧l 1_{and define}

the Hodge diﬀerential operatord _: _D_Ω_,_∧l 1 _→ _D_Ω_,_∧l_with_d ₋₁nl 1_d_{, where}

l0,1, . . . , n−1, andis the Hodge star operator. We denote a ball or cube byBand the ball

or cube with the same center asBand diamρB ρdiamBbyρB. The following nonlinear

elliptic equation:

dAx, dω Bx, dω 1.2

is called the nonhomogeneousA-harmonic equation of diﬀerential forms, where

A:Ω× ∧lRn−→ ∧lRn, B:Ω× ∧lRn−→ ∧l−1Rn 1.3

are operators satisfying the following conditions for almost allx∈Ωandξ∈ ∧l_Rn_:

|Ax, ξ|a|ξ|p−1,

|Bx, ξ|b|ξ|p−1, Ax, ξ, ξ ≥ |ξ|p,

1.4

wherea >0 andb > 0 are two constants, and 1 < p <∞is a fixed exponent dependent on

1.2.

If the operator B 0 in 1.2, it degenerates into the homogeneous A-harmonic

equation

dAx, dω 0. 1.5

The solutions to1.5are calledA-harmonic tensors. See 1 for the recent research onA

-harmonic equation.

2. A

β_r,λ

-Weighted Caccioppoli-Type Inequalities

Caccioppoli-type estimates have been widely studied and frequently used in analysis and

related fields, including partial diﬀerential equations and the theory of elasticity. These

inequalities provide upper bounds for the Lp_{-norm of} _∇u_if _u_{is a function or}_du_if _u_{is a}

form with theLp_{-norm of the di}_ﬀ_{erential form}_u_{. Di}_ﬀ_{erent versions of the Caccioppoli-type}

inequality have been established in2,3.

We first introduce the following definition of Aβ_r,λ-weights or the two-weight,

and then establish the localAβ_r,λ-weighted Caccioppoli-type inequality for solutions to the

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Definition 2.1. Assume that 1 ≤ r ≤ λ < ∞ and 0 ≤ β < n. One says a pair of weights

ω1x, ω2xsatisfies theAβ_r,λ-condition, writesω1x, ω2x∈Aβ_r,λΩ, if for all ballsB⊂

Ω, one has that

1

|B|1−β/n

B

ω1xdx

1/λ

B

ω2x1−rdx

1/r

C, for 1< r <∞, 1 r

1

r 1,

1

|B|1−β/n

B

ω1xdx

1/λ

ω2x, forr1 almost allx∈B,

2.1

whereC >0 is a constant.

In4, Nolder obtains the following local Caccioppoli-type estimate.

Lemma 2.2. Letu be anA-harmonic tensor inΩ and letσ > 1. Then there exists a constantC, independent ofuanddu, such that

dus,B ≤CdiamB−1u−cs,σB 2.2

for all balls or cubesBwithσB⊂Ωand all closed formsc. Here1< s <∞.

The next lemma is the generalized H ¨older inequality which will be widely used in this paper.

Lemma 2.3. Let0< α <∞,0< β <∞, ands−1_α−1 _β−1_{. If}_u_and_v_{are measurable functions on}

Rn_{, then}

uv_s,E ≤ uα,Evβ,E 2.3

for any measurable setE⊂Rn_.

The following weak reverse H ¨older inequality plays an important role in founding the

integral estimate of the nonhomogeneous and homogeneousA-harmonic tensor; see4.

Lemma 2.4. Letube a solution to1.2inΩ,σ >1and0< s,t <∞. Then there exists a constant

C, depending only ons, t, a, p, σ, andn, such that

us,B≤C|B|s−t/stut,σB 2.4

for all balls or cubesBwithσB⊂Ω.

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Lemma 2.5. EachΩhas a modified Whitney cover of cubesν{Qi}such that

i

Qi Ω,

Q∈ν

χ√₅_/₄_Q≤NχΩ

2.5

for allx ∈ Rn _{and some}_{N >} ₁_{, where}_χ

E is the characteristic function for a setE. Moreover, if

Qi∩Qj/∅, then there exists a cubeR(this cube does not need to be a member ofν) inQi∩Qjsuch

thatQi∪Qj⊂NR.

Now we are ready to prove the local weighted Caccioppoli-type inequality for

homogeneousA-harmonic tensors.

Theorem 2.6. Letu ∈Ls_Ω_,_∧l_,_l ₀_,₁_{, . . . , n}₋₁_{, be a solution to the homogeneous}_A_-harmonic

equation1.5andω1x, ω2x ∈Aβ_r,λΩfor some0 ≤β < nand1 ≤ r ≤ λ < ∞. Then there exists a constantC, independent ofuanddu, such that

B

|du|s

ωα₁dx

1/s

≤C|B|λα/s1/r−β/n−1/n−α/s σB

|u−c|s

ωλα/r₂ dx

1/s

2.6

for all balls or cubesBwithσB⊂Ωand all closed formsc. Specially, ifλr, one has

B

|du|s

ω₁αdx

1/s

≤C|B|−αβr/ns−1/n σB

|u−c|s

ωα₂dx

1/s

. 2.7

Hereαis any constant with0< α <1.

Proof. Choosingts/1−αand byLemma 2.3, we have

B

|du|s

ωα₁dx

1/s

B

|du|ωα/s

1 s

dx

1/s

≤

B

|du|t

dx

1/t

B

ω₁α/sst/t−sdx

t−s/st

du_t,B·

B

ω1dx t−s/st

.

2.8

Choosing 1< σ1 < σand byLemma 2.2, we know that

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wherecis a closed form. Putting2.9into2.8, we have

B

|du|s

ωα₁dx

1/s

≤ dut,B·

B

ω1dx t−s/st

≤C1diamB−1u−ct,σ1B

B

ω1dx α/s

.

2.10

Sincecis a closed form anduis a solution to1.5,u−cis still a solution to1.5.

Whenr >1, takingms/1 λ/rαr−1 rs/r λαr−1and byLemma 2.4, we obtain

u−ct,σ1B≤C2|B|

m−t/mt_u₋_c

m,σB. 2.11

ByLemma 2.3, we have

u−cm,σB

σB

|u−c|ωλα/rs

2 ω−

λα/rs 2 m dx 1/m ≤ σB

|u−c|s_ωλ/rα

2 dx 1/s σB 1 ω2

λαm/rs−m

dx

s−m/sm

u−c_s,σB,ωλ/rα 2 σB 1 ω2

1/r−1

dx

λα/s1−1/r

.

2.12

By the conditionω1x, ω2x∈Aβ_r,λΩ, we obtain

B

ω1dx

α/s

σB

1

ω2 1/r−1

dx

λα/s1−1/r

⎧ ⎨

⎩|B|1−β/n|B|β/n−1

B

ω1dx

1/λ

σB

1

ω2

1/r−1

dx

1−1/r⎫_⎬

⎭

λα/s

≤

⎧ ⎨

⎩|B|1−β/n|B|β/n−1

σB

ω1dx

1/λ

σB

1

ω2 1/r−1

dx

1−1/r⎫_⎬

⎭

λα/s

⎧ ⎨

⎩|B|1−β/n|B|β/n−1

σB

ω1dx

1/λ

σB

1

ω2 r−1

dx

1/r⎫_⎬

⎭

λα/s

≤C3|B|1−β/nλ/sα,

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where 1/r 1/r1. By2.10,2.11,2.12, and2.13, we conclude that

B

|du|s_ωα

1dx 1/s

≤C1diamB−1C2|B|m−t/mtu−cm,σB

B

ω1dx α/s

≤C4diamB−1|B|m−t/mtu−c_s,σB,ωλ/rα

2

B

ω1dx α/s

×

σB

1

ω2 1/r−1

dx

λα/s1−1/r

≤C5diamB−1|B|1/t−1/m|B|1−β/nλ/sαu−cs,σB,ωλα/r

2 .

2.14

Since diamB C6|B|1/n, inequality2.14can be rewritten as

B

|du|s

ωα₁dx

1/s

C7|B|−1/n|B|1/t−1/m|B|1−β/nλ/sαu−c_s,σB,ωλα/r

2 . 2.15

By simple computation, we know that

1

t −

1

m

1−α

s −

1 λα/rr−1

s − α s −

λα s

1−1

r

. 2.16

Then we can change inequality2.15into

B

|du|s

ωα1dx 1/s

≤C|B|λα/s1/r−β/n−1/n−α/s_u₋_c s,σB,ωλα/r

2 . 2.17

Whenr1, takingm < sand byLemma 2.4, we obtain

u−ct,σ1B≤C8|B|

m−t/mt_u₋_c

m,σB. 2.18

ByLemma 2.3, it follows that

u−cm,σB

σB

|u−c|ωλα/s

2 ω−2λα/smdx 1/m

≤

σB

|u−c|s_ωλα

2 dx

1/s

σB

1

ω2

λαm/s−m

dx

s−m/sm

u−cs,σB,ωλα

2

|σB|1−β/n

σB

ω1xdx

−1/λλα/s

|σB|s−m/sm

u−c_s,σB,ωλα

2 |σB|

λα/s1−β/n

σB

ω1xdx

₋α/s

|σB|s−m/sm_.

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By2.10,2.18, and2.19, we conclude that

B

|du|s_ωα

1dx 1/s

≤C1diamB−1C8|B|m−t/mtu−cm,σB

B

ω1dx α/s

≤C9diamB−1|B|m−t/mtu−cs,σB,ωλα

2

B

ω1dx α/s

× |σB|λα/s1−β/n

σB

ω1xdx

₋α/s

|σB|s−m/sm

≤C10diamB−1|B|1/t−1/m|σB|λα/s1−β/n|σB|1/m−1/su−cs,σB,ωλα

2 . 2.20

Since diamB C11|B|1/nand|σB|σn|B|, inequality2.20can be rewritten as

B

|du|s_ωα

1dx 1/s

≤C12|B|−1/n|B|1/t−1/m|B|1−β/nλα/s|B|1/m−1/su−cs,σB,ωλα

2

C12|B|−1/n 1/t−1/s 1−β/nλα/su−cs,σB,ωλα

2 .

2.21

Because ofts/1−α, we have that

B

|du|s_ωα

1dx 1/s

≤C|B|λα/s1−β/n−1/n−α/s_u₋_c s,σB,ωλα

2 . 2.22

The proof ofTheorem 2.6is completed.

Since there are fore real parameters,α, λ, β, andr, inTheorem 2.6, we can obtain some

desired versions of weighted Caccioppoli-type estimates by diﬀerent choices of them. Let

βn−1 inTheorem 2.6, we obtain the following corollaries.

Corollary 2.7. Letu∈Ls_Ω_,_∧l_,_l ₀_,₁_{, . . . , n}₋₁_,_{be a solution to the homogeneous}_A_-harmonic

equation1.5andω1x, ω2x∈An−1

r,r Ωfor some0< α <1and1< r <∞. Then there exists a

constantC, independent ofuanddu, such that

B

|du|s

ωα₁dx

1/s

≤C|B|αr/ns−1/n−αr/s

σB

|u−c|s

ωα2dx 1/s

2.23

for all closed formsc.

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Corollary 2.8. Letu∈Ls_Ω_,_∧l_,_l ₀_,₁_{, . . . , n}₋₁_,_{be a solution to the homogeneous}_A_-harmonic

equation1.5andω1x, ω2x∈An−1

r,r Ωfor some1 < r <∞. Then there exists a constantC,

independent ofuanddu, such that

B

|du|s_ω1/r

1 dx 1/s

≤C|B|1/ns−1/n−1/s

σB

|u−c|s_ω1/r

2 dx 1/s

2.24

for all closed formsc.

3. The Local Weighted Estimates for the Decomposition

The Hodge decomposition theorem has been playing an important part in partial diﬀerential

equation, the operator theory, and so on. In the recent years, there are some interesting conclusions on the Hodge decomposition of diﬀerential forms; see5,6. In7, there is the following decomposition theorem for diﬀerential formu∈Lp_Rn_,_∧l_.

Lemma 3.1. Letu∈Lp_Rn_,_∧l_,₁ _{< p <}_∞_,_l ₁_,₂_{, . . . , n}₋₁_{. Then, there exist di}_ﬀ_{erential forms}

Q∈LP₁Rn_,_∧l−1_{and R}_∈_LP

1Rn,∧l 1, such that

udQ d_R, _d_Q_dR₀_, _3.1

∇Q_p ∇R_p≤Cu_p, 3.2

hereCis a positive constant.

Definition 3.2. The weightωxsatisfies theAr-condition on the setE⊂Rn, writeω∈ArE,

ifωx>0, a.e., and for all ballsB⊂E, one has that

sup

B

1

|B|

B

ωdx

1

|B|

B

1

ω

1/r−1

dx

r−1

<∞, 3.3

wherer >1.

The next lemma states the reverse H ¨older inequality forArE-weight; see8.

Lemma 3.3. Assume thatω∈ArE. Then, there exists a constantC, independent ofω, for all balls

B⊂E, such that

ωβ,B≤C|B|1−β/βω1,B, 3.4

whereβ >1is a real number.

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Theorem 3.4. Letu∈Lp_Ω_,_∧l_{be the solution to the nonhomogeneous}_A_{-harmonic equation}_1.2_,

1 < p < ∞,l 1,2, . . . , n−1. Then, there exist diﬀerential forms Q ∈ LP

1Ω,∧l−1 and R ∈

LP

1Ω,∧l 1, such that

udQ dR, dQdR0, 3.5

∇Qp,B,ωα ∇R_p,B,ωα ≤Cu_p,σB,ωα, 3.6

hereω∈ArΩ,r >1,σ >1,0< α≤1, andC >0is a constant.

Proof. From3.2, it follows that

∇Qs≤C1us, 3.7

∇Rs≤C2us 3.8

for anys >1.

When 0< α <1, takesp/1−α> p >1. ByLemma 2.3with 1/p1/s s−p/sp

and3.7, we have

∇Qp,B,ωα

B

|∇Q|p

ωαdx

1/p

B

|∇Q|ωα/pp_dx

1/p

≤

B

|∇Q|s

dx

1/s

B

ωα/psp/s−pdx

s−p/sp

∇Qs,B

B

ωdx

α/p

≤C1us,B

B

ωdx

α/p

.

3.9

Similarly, usingLemma 2.3and3.8, we have

∇Rp,B,ωα

B

|∇R|p

ωαdx

1/p

B

|∇R|ωα/pp_dx

1/p

≤

B

|∇R|s_dx

1/s

B

ωα/psp/s−pdx

s−p/sp

∇Rs,B

B

ωdx

α/p

≤C2us,B

B

ωdx

α/p

.

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Combining3.9with3.10, we have

∇Qp,B,ωα ∇R_p,B,ωα ≤C1us,B

B

ωdx

α/p

C2us,B

B

ωdx

α/p

≤C3us,B

B

ωdx

α/p

.

3.11

Taketp/1 αr−1, thent < p < sandp−t/ptαr−1/p. ByLemma 2.4, it follows that

us,B≤C4|B|t−s/stut,σB, 3.12

hereσ >1 is a constant. Putting3.11into3.12, we get

∇Q_p,B,ωα ∇R_p,B,ωα ≤C3us,B

B

ωdx

α/p

≤C5|B|t−s/stut,σB

B

ωdx

α/p

.

3.13

ByLemma 2.3, the following inequality holds:

ut,σB

σB

|u|t

dx

1/t

σB

|u|ωα/p_ω−α/pt

dx

1/t

≤

σB

|u|p

ωαdx

1/p

σB

ω−α/ppt/p−tdx

p−t/pt

σB

|u|p

ωαdx

1/p

σB

1

ω

1/r−1

dx

αr−1/p

u_p,σB,ωα

σB

1

ω

1/r−1

dx

αr−1/p

.

3.14

Combining3.13with3.14, we obtain the following estimate:

∇Qp,B,ωα ∇R_p,B,ωα ≤C5|B|t−s/stut,σB

B

ωdx

α/p

≤C5|B|t−s/stup,σB,ωα

B

ωdx

α/p

·

σB

1

ω

1/r−1

dx

αr−1/p

.

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Sinceω∈ArΩ, we have

B

ωdx

α/p

·

σB

1

ω

1/r−1

dx

αr−1/p

≤

σB

ωdx

α/p

·

σB

1

ω

1/r−1

dx

αr−1/p

⎧ ⎨ ⎩|σB|r

1

|σB|

σB

ωdx

1

|σB|

σB

1

ω

1/r−1

dx

r−1⎫_⎬

⎭

α/p

≤C6|B|αr/p.

3.16

Putting3.16into3.15, we get

∇Qp,B,ωα ∇R_p,B,ωα≤C₇|B|t−s/st|B|αr/pu_p,σB,ωα. 3.17

Recall thatsp/1−α,tp/1 αr−1, then we have

t−s ts

p1−α−p1 αr−1

p2 −

αr

p . 3.18

So3.17can be rewritten as

∇Qp,B,ωα ∇R_p,B,ωα ≤C7up,σB,ωα. 3.19

Whenα1, byLemma 3.3, we know that

ωβ,B≤C8|B|1−β/βω1,B, 3.20

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Let s βp/β−1, then 1 < p < s and β s/s−p. ByLemma 2.3with 1/p

1/s s−p/spand3.20, we have that

∇Q_p,B,ω

B

|∇Q|p_ωdx

1/p

B

|∇Q|ω1/pp_dx

1/p

≤

B

|∇Q|s

dx

1/s

B

ω1/psp/s−pdx

s−p/sp

∇Qs,B

B

ωβdx

1/pβ

≤C9|B|1−β/pβω11/p,B∇Qs,B.

3.21

Similarly, byLemma 2.3and3.20, we have that

∇Rp,B,ω

B

|∇R|p

ωdx

1/p

B

|∇R|ω1/pp_dx

1/p

≤

B

|∇R|s_dx

1/s

B

ω1/psp/s−pdx

s−p/sp

∇Rs,B

B

ωβdx

1/pβ

≤C10|B|1−β/pβω1₁/p_,B∇Rs,B.

3.22

Putting3.7into3.21, we have

∇Qp,B,ω≤C9|B|1−β/pβω1₁/p_,B∇Qs,B

≤C11|B|1−β/pβω1₁/p_,Bus,B.

3.23

Putting3.8into3.22, we conclude that

∇Rp,B,ω≤C10|B|1−β/pβω1₁/p_,B∇Rs,B

≤C12|B|1−β/pβω1₁/p_,Bus,B.

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Combining3.23with3.24, we obtain

∇Qp,B,ω ∇Rp,B,ω

≤C11|B|1−β/pβω1₁/p_,Bus,B C12|B|1−β/pβω1₁/p_,Bus,B

≤C13|B|1−β/pβω1₁/p_,Bus,B.

3.25

Lettp/r, thent < p. ByLemma 2.4, we have

us,B ≤C14|B|t−s/stut,σB, 3.26

whereσ >1. Putting3.26into3.25, we get

∇Q_p,B,ω ∇R_p,B,ω≤C13|B|1−β/pβω1₁/p_,Bus,B

≤C15|B|1−β/pβ|B|t−s/stut,σBω

1/p

1,B.

3.27

ByLemma 2.3, we get

u_t,σB

σB

|u|t_dx

1/t

σB

|u|ω1/p_ω−1/pt_dx

1/t

≤

σB

|u|p_ωdx

1/p

σB

1

ω

pt/p−t

dx

p−t/pt

up,σB,ω

σB

1

ω

1/r−1

dx

r−1/p

.

3.28

Putting3.28into3.27, it follows that

∇Qp,B,ω ∇Rp,B,ω

≤C15|B|1−β/pβ|B|t−s/stut,σBω

1/p

1,B

≤C15|B|1−β/pβ|B|t−s/stup,σB,ω

σB

1

ω

1/r−1

dx

r−1/p

B

ωdx

1/p

.

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Sinceω∈ArΩ, we have

B

ωdx

1/p

·

σB

1

ω

1/r−1

dx

r−1/p

≤

σB

ωdx

1/p

·

σB

1

ω

1/r−1

dx

r−1/p

⎧ ⎨ ⎩|σB|r

1

|σB|

σB

ωdx

1

|σB|

σB

1

ω

1/r−1

dx

r−1⎫_⎬

⎭ 1/p

≤C16|B|r/p.

3.30

Putting3.30into3.29, we get the following inequality:

∇Qp,B,ω ∇Rp,B,ω≤C17|B|1−β/pβ|B|t−s/st|B|r/pup,σB,ω. 3.31

Since1−β/pβ t−s/st r/p−1/s t−s/st 1/t t−s/st s−t/st0,3.31

can be rewritten as

∇Qp,B,ω ∇Rp,B,ω≤C17up,σB,ω. 3.32

So inequality3.4is true for 0< α≤1.

4. The Global Weighted Estimates

Based on the local weighted estimate for the decomposition and the Whitney covering

lemma, we get the global weighted estimate on domainΩ⊂Rn_.

Theorem 4.1. Letu∈Lp_Ω_,_∧l_{be a di}_ﬀ_{erential form satisfying the nonhomogeneous}_A_-harmonic

equation1.2in a bounded domainΩ⊂Rn_,₁ _{< p <}_∞_,_l₁_,₂_{, . . . , n}₋₁_{, and}_ω_∈_A

rΩ,r >1.

Then, there exist diﬀerential formsQ∈LP

1Ω,∧l−1andR∈LP1Ω,∧l 1, such that

udQ dR, dQdR0,

∇Qp,Ω,ωα ∇R_p,_Ω_,ωα ≤Cu_p,_Ω_,ωα,

4.1

(15)

Proof. By Lemma 2.5, Ω ⊂ Rn _{has a modified Whitney cover of cubes} _ν _{B i}. Let

σ5/4.

First, we assume that all cubesBisatisfyσBi ⊂ Ω. ByLemma 2.5and3.6, we know

that

∇Qp,Ω,ωα ∇R_p,_Ω_,ωα

Ω|∇Q|

p

ωαdx

1/p

Ω|∇R|

p

ωαdx

1/p

≤

B∈ν

B

|∇Q|p_ωα_dx

1/p

B

|∇R|p_ωα_dx

1/p

≤

B∈ν

C1

σB

|u|p

ωαdx

1/p

≤

B∈ν

C2

σB

|u|p_ωα_dx

1/p

χσBx

≤

B∈ν

C2

Ω|u|

p

ωαdx

1/p

χσBx

≤

Ω|u|

p_ωα_dx

1/p

B∈ν

C2χσBx

≤C

Ω|u|

p

ωαdx

1/p

.

4.2

In the above proof, if there exists one cubeB0∈ν, such thatσB0can not be contained

in Ω completely, the proof is as follows. Set Ω1 _B_∈_νσB, and define Ux on Ω1 as

follows:

Ux

⎧ ⎪ ⎨ ⎪ ⎩

ux, x∈Ω,

0 x∈Ω1−Ω.

4.3

Then, we know that the following formulas are true:

σB

|u|p

ωαdx

1/p

σB

|U|p

ωαdx

1/p

,

Ω1

|U|p

ωαdx

1/p

Ω|u|

p

ωαdx

1/p

.

(16)

Note that4.4hold, and then the following formula is true:

∇Qp,Ω,ωα ∇R_p,_Ω_,ωα

Ω|∇Q|

p

ωαdx

1/p

Ω|∇R|

p

ωαdx

1/p

≤

B∈ν

B

|∇Q|p_ωα_dx

1/p

B

|∇R|p_ωα_dx

1/p

≤

B∈ν

C3

σB

|u|p

ωαdx

1/p

B∈ν

C3

σB

|U|p

ωαdx

1/p

≤

B∈ν

C4

σB

|U|p

ωαdx

1/p

χσBx

≤

B∈ν

C5

Ω1

|U|p

ωαdx

1/p

χσBx

≤

Ω1

|U|p_ωα_dx

1/p

B∈ν

C5χσBx

≤C

Ω1

|U|p

ωαdx

1/p

C

Ω|u|

p_ωα_dx

1/p

.

4.5

The proof ofTheorem 4.1is completed.

In Theorems3.4and 4.1, there is a real parameter α, which makes the results more

flexible. By choosing diﬀerent value of the parameter α, we get the estimates in diﬀerent

forms. For example, if we takeα1, we have the following corollary.

Corollary 4.2. Letu∈Lp_Ω_,_∧l_{be the solution to the nonhomogeneous}_A_{-harmonic equation}_1.2_,

1 < p < ∞,l 1,2, . . . , n−1, and ω ∈ ArΩ,r > 1. Then, there exist diﬀerential formsQ ∈

LP

1Ω,∧l−1andR∈LP1Ω,∧l 1, such that

udQ dR, dQdR0,

∇Qp,Ω,ω ∇Rp,Ω,ω≤Cup,Ω,ω,

4.6

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Acknowledgments

This work was supported by the National Natural Science Foundation of ChinaGrant no.

10771044and the Natural Science Foundation of Heilongjiang ProvinceGrant no. 200605.

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