(λ) -WEIGHTED CACCIOPPOLI-TYPE AND POINCARÉ-TYPE INEQUALITIES FOR A -HARMONIC TENSORS

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A

(λ) -WEIGHTED CACCIOPPOLI-TYPE AND POINCARÉ-TYPE INEQUALITIES FOR A -HARMONIC TENSORS

BING LIU Received 16 July 2001

We prove a local version of weighted Caccioppoli-type inequality, then we prove a version of weighted Poincaré-type inequality forA-harmonic tensors both locally and globally.

2000 Mathematics Subject Classification: 30C65, 31B05, 58A10.

1. Introduction. There have been many studies for the integrability of differential forms, and the estimations of the integrals of differential forms. As extensions of those studies, the integrability and estimations of integrals forA-harmonic tensors are also studied and applied in many fields such as in tensor analysis, potential theory, partial differential equations, and quasiregular mappings, see [1, 2, 6, 7, 8, 9, 10, 11, 12].

There are many studies about Caccioppoli-type and Poincaré-type inequalities forA- harmonic tensors, see [3,4,5,12]. We state the following specific results from [12].

Theorem1.1. Letube anA-harmonic tensor inΩand letσ >1. Then there exists a constantC, independent ofuanddu, such that

dus,B≤C|B|⁻¹u−cs,σ B (1.1)

for all balls or cubesBwithσ B⊂Ω.

Theorem1.2. Letu∈D(Q,∧^l)anddu∈L^p(Q,∧^l+1). Then there exists a closed formuQ, defined inQ, such thatu−uQis inW_p¹(Q,∧^l)with1< p <∞and

u−uQ_p,Q≤C(n, p)|Q|^1/ndup,Q (1.2)

forQa cube or a ball inRⁿ,l=0,1, . . . , n.

Our work is to give new versions of Theorems1.1and1.2withAr(λ)weight. When λ=1, the properties ofAr(1)-weight can be found in [6].

We first introduce some related definitions and notations which are adopted from [12].

We assume thatΩis a connected open subset ofRⁿ. The Lebesgue measure of a setE⊂Rⁿis denoted by|E|. Balls inRⁿare denoted byBandσ Bis the ball with the same center asBand with diam(σ B)=σdiam(B). We callwa weight ifw∈L¹_loc(Rⁿ) andw >0 a.e.

Lete1, e2, . . . , en be the standard unit basis ofRⁿ. Assume that∧^l= ∧^l(Rⁿ)is the linear space ofl-vectors, spanned by the exterior products eI=ei₁∧ei₂∧ ··· ∧ei_l

corresponding to all orderedl-tuplesI=(i1, i2, . . . , il), 1≤i1< i2<···< il≤n. The Grassman algebra∧ = ⊕∧^lis a graded algebra with respect to the exterior products.

Forα=

α^IeI∈ ∧and β=

β^IeI∈ ∧, the inner product in∧is given by α, β = α^Iβ^I with summation over alll-tuplesI=(i1, . . . , il)and all integersl=0,1, . . . , n.

The Hodge star operator∗:∧ → ∧is defined by∗1=e1∧e2∧···∧enandα∧∗β= β∧ ∗α= α, β(∗1)for allα, β∈ ∧. The norm ofα∈ ∧is given by|α|²= α, α =

∗(α∧∗α)∈ ∧⁰=R. The Hodge star is an isometric isomorphism on∧with∗:∧^l→

∧^n−land∗∗(−1)^l(n−l):∧^l→ ∧^l.

For 0≤k≤n, ak-formω(x)∈ ∧^k(R)ⁿis defined by

ω(x)=

I

ωI(x)dxI=

ωi₁,i₂,...,i_k(x)dxi₁∧dxi₂∧···∧dxi_k, (1.3)

whereωi₁,i₂,...,i_k(x)are real functions inRⁿ,I =(i1, i2, . . . , ik),ij∈ {1,2, . . . , n}and j =1,2, . . . , k. The function ω(x)is called a differential k-form ifωi₁,i₂,...,i_k(x)are differentiable functions. Note that a differential 0-form is a differentiable function f : Rⁿ →R. A differential l-form ω onΩ is a locally integrable function or more generally, a Schwartz distribution onΩwith values in∧^l(Rⁿ). We denoteD(Ω,∧^l)as a space of all differentiall-forms andL^p(Ω,∧^l)as a space of differentiall-forms with coefficients in theL^p(Ω,Rⁿ). The spaceL^p(Ω,∧^l)is a Banach space with the norm

ωp,Ω=

Ω

ω(x)^pdx 1/p

=

Ω

I

ωI(x)²p/2

dx 1/p

. (1.4)

We also denoteW_p¹(Ω,∧^l)as a space of differentiall-forms onΩwhose coefficients are in Sobolev spaceW_p¹(Ω,R).

AnA-harmonic equation for differential forms is

dA(x, dω)=0, (1.5)

whered:D(Ω,∧^l+1)→D(Ω,∧^l), as the formal adjoint operator ofd, is given by

d=(−1)^nl+1 d (1.6)

on D(Ω,∧^l+1), l= 0,1, . . . , n, and A: Ω× ∧^l(Rⁿ)→ ∧^l(Rⁿ) satisfies the following conditions:

A(x, ξ)≤a|ξ|^p−1,

A(x, ξ), ξ ≥ |ξ|^p (1.7)

for almost everyx∈Ωand allξ∈ ∧^l(Rⁿ). Herea >0 is a constant and 1< p <∞is a fixed exponent associated with (1.5). LetW_p,loc¹ (Ω,∧^l−1)= ∩W_p¹(Ω,∧^l−1), where the

r

intersection is for allΩcompactly contained inΩ. A solution to (1.5) is an element of the Sobolev spaceW_p,loc¹ (Ω,∧^l−1)such that

Ω

A(x, dω), dϕ =0 (1.8)

for allϕ∈W_p¹(Ω,∧^l−1)with compact support, see [8,9,12].

Definition1.3. We calluanA-harmonic tensor inΩifusatisfies theA-harmonic equation (1.5) inΩ.

The following definition belongs to Ding and Shi [5].

Definition1.4. The weightw(x)satisfies theAr(λ)condition,r >1,λ >1, write w∈Ar(λ), ifw(x) >0 a.e. and

sup

B

1

|B|

Bw^λdx 1

|B|

B

1 w

1/(r−1)

dx (r−1)

<∞ (1.9)

for any ballB⊂Rⁿ.

The following lemma is from Nolder [12].

Lemma1.5. EachΩhas a modified Whitney cover of cubesW= {Qi}which satisfy

∪Qi=Ω,

Q∈W

χ√

5/4Q≤Nχ_Ω (1.10)

for allx∈Rⁿand someN >1and ifQi∩Qj≠∅, then there exists a cubeR (∈W )in Qi∩Qjsuch thatQi∪Qj⊂NR.

We also need the following generalized Hölder’s inequality.

Lemma1.6. Let0< α <∞,0< β <∞, ands⁻¹=α⁻¹+β⁻¹. Iffandgare measur- able functions onRⁿ, then

f gs,Ω≤ fα,Ωgβ,Ω (1.11)

for anyΩ⊂Rⁿ.

2. Main results

Theorem2.1. Letu∈D(Ω,∧^l),l=0,1, . . . , n, be anA-harmonic tensor in a do- mainΩ⊂Rⁿ andρ >1. Assume that1< s <∞ is a fixed exponent associated with theA-harmonic equation and weightw∈Ar(λ)for somer >1andλ >0. Then there exists a constantC, independent ofuanddu, such that

B|du|^sw^sλ/(1+s)dx 1/s

≤ C

|B|

ρB|u−c|^sw^s/(1+s)dx 1/s

(2.1)

for all ballsBwithρB⊂Ωand all closed formsc.

Proof. Chooset=s(s+1)for given 1< s <∞, then 1< s < t. Applying Hölder’s inequality (1.11) andTheorem 1.1, we obtain

B|du|^sw^sλ/(s+1)dx 1/s

=

B

|du|w^λ/(s+1)s

dx 1/s

≤

B|du|^tdx 1/t

B

w^λ/(s+1)st/(t−s)

dx

(t−s)/(st)

≤C1|B|⁻¹u−ct,ρB

Bw^λdx 1/(s+1)

≤C1|B|⁻¹

B

w^λdx 1/(s+1)

|ρB|^(m−t)/mtu−cm,ρB, (2.2)

where the last inequality was obtained by choosingm=(s+s²)/(sr+1)and applying (1.11) tou−ct,ρBwith 1/t=1/m+(m−t)/mt.

Sincem > s, using Hölder’s inequality, we have

ρB|u−c|^mdx 1/m

=

ρB

|u−c|w^1/(s+1)w^−1/(s+1)m

dx 1/m

≤

ρB

|u−c|w^1/(s+1)s

dx 1/s

ρB

w^−1/(s+1)ms/(s−m)

dx

(s−m)/ms

. (2.3)

By the choice ofm,(s−m)(s+1)/ms=r−1. Sincew∈Ar(λ), we have

B

w^λdx

1/(s+1)

ρB

1 w

ms/(s−m)(s+1)

dx

(s−m)(s+1)/(ms(s+1))

=





B

w^λdx

ρB

1 w

1/(r−1)

dx r−1



1/(s+1)

≤



|ρB|^r 1

|ρB|

ρBw^λdx 1

|ρB|

ρB

1 w

1/(r−1)

dx r−1



1/(s+1)

≤C2|ρB|^{r /(s+1)}.

(2.4)

r

Thus, putting (2.2), (2.3), and (2.4) together and noting that(m−t)/mt= −r /(s+1), we have

B|du|^sw^sλ/(s+1)dx 1/s

≤C1|B|⁻¹|ρB|^(m−t)/mtC2|ρB|^{r /(s+1)}

ρB|u−c|^sw^s/(s+1)dx 1/s

≤C3|B|⁻¹

ρB|u−c|^sw^s/(s+1)dx 1/s

.

(2.5)

We have provedTheorem 2.1.

By choosing different values ofλinTheorem 2.1, we get different versions of the Caccioppoli-type inequality. For example, ifλ=(s+1)/s,(2.1) reduces to

B|du|^sw dx 1/s

≤ C

|B|

ρB|u−c|^sw^s/(s+1)dx 1/s

. (2.6)

If we chooseλ=s+1 inTheorem 2.1, (2.1) is in the form of

B|du|^sw^sdx 1/s

≤ C

|B|

ρB|u−c|^sw^s/(s+1)dx 1/s

. (2.7)

Ifλ=1, we get the symmetric form of (2.1),

B|du|^sw^s/(s+1)dx 1/s

≤ C

|B|

ρB|u−c|^sw^s/(s+1)dx 1/s

. (2.8)

Now we consider a type of Poincaré inequality withAr(λ)-weights. The following is a local result for any connected open subset inRⁿ.

Theorem2.2. Letu∈D(Ω,∧^l)be anA-harmonic tensor in a domainΩ⊂Rⁿand du∈L^s(Ω,∧^l+1),l=0,1, . . . , n. Assume thatσ >1,1< s <∞, andw∈Ar(λ)for some r >1and any real numberλ >0. Then

1

|B|

B

u−uB^sw^λ/sdx 1/s

≤C|B|^1/n 1

|B|

σ B|du|^sw^1/sdx 1/s

(2.9)

for all ballsBwithσ B⊂Ω. HereCis a constant independent ofu.

Proof. We only need to prove the following:

B

u−uB^sw^λ/sdx 1/s

≤C|B|^1/n

σ B|du|^sw^1/sdx 1/s

. (2.10)

For any 1< s <∞, chooset such that t=s²/(s−1), then 1< s < t. By Hölder’s inequality we have

B

u−uB^sw^λ/sdx 1/s

=

B

u−uBw^λ/s2s

dx 1/s

≤

B

u−uB^tdx 1/t

B

w^λ/s2st/(t−s)

dx (t−s)/st

=

B

u−uB^tdx 1/t

Bw^λdx 1/s²

.

(2.11)

Using Hölder’s inequality again and choosingmsuch thatm=s²/(s+r−1)for any givenr >1, we obtain

B

u−uB^tdx 1/t

≤ |σ B|^(m−t)/(mt)

σ B

u−uB^mdx 1/m

(2.12)

for anyσ >1.Thus, byTheorem 1.2,

B

u−uB^tdx 1/t

≤ |σ B|^(m−t)/mtC(n, m)|σ B|^1/n

σ B|du|^mdx 1/m

, (2.13)

where

σ B|du|^mdx 1/m

=

σ B|du|^mw^1/s2w^−1/s2dx 1/m

≤

σ B

|du|w^1/s2s

dx 1/s

σ B

w^−1/s2ms/(s−m)

dx

(s−m)/ms

=

σ B|du|^sw^1/sdx 1/s

σ B

1 w

1/(r−1)

dx

(r−1)/s²

.

(2.14)

Substituting (2.12), (2.13), and (2.14) to inequality (2.11), and usingAr(λ)condition tow, we have

B

u−uB^sw^λ/sdx 1/s

≤C(n, m)|σ B|^1/n

Bw^λdx 1/s²

|σ B|^(m−t)/mt

σ B|du|^sw^1/sdx 1/s

×

σ B

1 w

1/(r−1)

dx

(r−1)/s²

r

≤C(n, m)|σ B|^1/n

σ B|du|^sw^1/sdx 1/s

×



 1

|σ B|

σ B

w^λdx 1

|σ B|

σ B

1 w

1/(r−1)

dx r−1



1/s²

≤C1(n, m)|B|^1/n

σ B|du|^sw^1/sdx 1/s

.

(2.15)

Theorem2.3(global result ofTheorem 2.2). Letu∈D(Ω,∧^l)be anA-harmonic tensor in a domainΩ⊂Rⁿanddu∈L^s(Ω,∧^l+1),l=0,1, . . . , n. Assume that1< s <∞ andw∈Ar(λ)for somer >1andλ >0. Then

1

|Ω|

Ω

u−uB^sw^λ/sdx 1/s

≤C|Ω|^1/n 1

|Ω|

Ω|du|^sw^1/sdx 1/s

. (2.16)

Proof. ByLemma 1.5, there exists a Whitney coverF= {Qi}ofΩ. In particular, we can choose 1< σ≤

5/4 inTheorem 2.2, so that

Ω

u−uB^sw^λ/sdx≤

Q∈FQ

u−uB^sw^λ/sdx

≤

Q∈F

Q

u−uB^sw^λ/sdx

≤

Q∈F

C1|Q|^s/n

σ Q|du|^sw^1/sdx

≤C2|Ω|^s/n

Q∈F

σ Q|du|^sw^1/sdxχσ Q

≤C3|Ω|^s/n

Q∈F

Ω|du|^sw^1/sdxχ√

5/4Q

≤C4|Ω|^s/n

Ω|du|^sw^1/sdx

Q∈F

χ√

5/4Q

≤C5|Ω|^s/n

Ω|du|^sw^1/sdxNχ_Ω(x)

≤C6|Ω|^s/n

Ω|du|^sw^1/sdx.

(2.17)

Thus,

1

|Ω|

Ω

u−uB^sw^λ/sdx≤C6

1

|Ω||Ω|^s/n

Ω|du|^sw^1/sdx (2.18) or

1

|Ω|

Ω

u−uB^sw^λ/sdx 1/s

≤C6|Ω|^1/n 1

|Ω|

Ω|du|^sw^1/sdx 1/s

. (2.19)

Remark2.4. Similar toTheorem 2.1, we have different versions of global results for Poincaré-type inequality by choosing different values ofλ. For instance, asλ=1, (2.16) reduces to

1

|Ω|

Ω

u−uB^sw^1/sdx 1/s

≤C|Ω|^1/n 1

|Ω|

Ω|du|^sw^1/sdx 1/s

. (2.20)

Asλ=s, (2.16) is in the form of 1

|Ω|

Ω

u−uB^sw dx 1/s

≤C|Ω|^1/n 1

|Ω|

Ω|du|^sw^1/sdx 1/s

. (2.21)

And ifλ=s²in (2.16), we have 1

|Ω|

Ω

u−uB^sw^sdx 1/s

≤C|Ω|^1/n 1

|Ω|

Ω|du|^sw^1/sdx 1/s

. (2.22)

Since parameterλ >0 can be chosen arbitrarily, the inequalities in our theorems can be used to estimate a relatively broad class of integrals.

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Bing Liu: Department of Mathematical Sciences, Saginaw Valley State University, University Center, MI48710, USA

E-mail address:[email protected]