Volume 2008, Article ID 397340,15pages doi:10.1155/2008/397340
Research Article
The Reverse H ¨older Inequality for
the Solution to
p
-Harmonic Type System
Zhenhua Cao,1Gejun Bao,1 Ronglu Li,1and Haijing Zhu2
1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2College of Mathematics and Physics, Shan Dong Institute of Light Industry, Jinan 250353, China
Correspondence should be addressed to Gejun Bao,[email protected]
Received 6 July 2008; Revised 9 September 2008; Accepted 5 November 2008
Recommended by Shusen Ding
Some inequalities toA-harmonic equationAx, du d∗v have been proved. TheA-harmonic equation is a particular form ofp-harmonic type systemAx, adu bd∗vonly whena0 andb0. In this paper, we will prove the Poincar´e inequality and the reverse H ¨older inequality for the solution to thep-harmonic type system.
Copyrightq2008 Zhenhua Cao et al. 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
Recently, amount of work about theA-harmonic equation for the differential forms has been done. In fact, the A-harmonic equation is an important generalization of the p-harmonic equation inRn,p >1, and thep-harmonic equation is a natural extension of the usual Laplace equationsee1for 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 elasticitysee2. In 1999, Nolder gave the reverse H ¨older inequality for the solution to theA-harmonic equation in3, and different versions of Caccioppoli estimates have been established in4–6. In 2004, D’Onofrio and Iwaniec introduced thep-harmonic type system in7, which is an important extension of the conjugateA-harmonic equation. In 2007, Ding proved the following inequality in8.
Theorem A. Letu, vbe a pair of solutions toAx, gdu hd∗vin a domainΩ ⊂ Rn. If
g ∈LpB,ΛLandh∈LqB,ΛL, thendu∈LpB,ΛLif and only ifd∗v∈LqB,ΛL. Moreover, there exist constantsC1, C2independent ofuandv, such that
d∗vqq,B≤C1
hqq,Bgpp,Bdupp,B,
dupp,B≤C2
In this paper, we will prove the Poincar´e inequalityseeTheorem 2.5and the reverse H ¨older inequality for the solution to thep-harmonic type systemseeTheorem 3.5. Now let us see some notions and definitions about thep-harmonic type system.
Let e1, e2, . . . , en denote the standard orthogonal basis ofRn. For l 0,1, . . . , n,we
denote byΛl ΛlRnthe 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.
The Grassmann algebraΛ ⊕Λlis a graded algebra with respect to the exterior products. For
ααIeI∈ΛandββIeI ∈Λ, then its inner product is obtained by
α, βαIβI, 1.2
with the summation over allI i1, i2, . . . , iland all integersl 0,1, . . . , n. The Hodge star
operator∗:Λ → Λis defined by the rule
∗1ei1∧ei2∧ · · · ∧ein,
α∧ ∗ββ∧ ∗αα, β∗1 ∀α, β∈Λ. 1.3
Hence, the norm ofα∈Λcan be given by
|α|2α, α∗α∧ ∗α∈Λ
0R. 1.4
Throughout this paper,Ω⊂ Rn is an open subset, for any constantσ > 1,Qdenotes a cube such thatQ ⊂ σQ ⊂ Ω, where σQdenotes the cube whose center is as same asQ and diamσQ σdiamQ. We sayα αIeI ∈ Λ is a differentiall-form on Ω, if every
coefficientαI of αis Schwartz distribution onΩ. The space spanned by differentiall-form
on Ω is denoted by DΩ,Λl. We write LpΩ,Λl for the l-form α α
IdxI onΩ with
αI ∈LpΩfor all orderedl-tupleI. ThusLpΩ,Λlis a Banach space with the norm
αp,Ω
Ω|α|
pdx 1/p
Ω
I
|αI|2
p/2
dx
1/p
. 1.5
SimilarlyWk,pΩ,Λldenotes thosel-forms onΩwith all coefficients inWk,pΩ. We denote
the exterior derivative by
d:DΩ,Λl−→DΩ,Λl1, forl0,1,2, . . . , n, 1.6
and its formal adjoint operatorthe Hodge codifferential operator
d∗:DΩ,Λl−→DΩ,Λl−1. 1.7
The operatorsdandd∗are given by the formulas
dα
I
The following two definitions appear in7.
Definition 1.1. The Hodge system holds:
Ax, adu bd∗v, 1.9
wherea∈LpΩ,Λlandb∈LqΩ,Λl, is ap-harmonic type system ifAis a mapping from
Ω×ΛltoΛlsatisfying
1x → Ax, ξis measurable inx∈Ωfor everyξ∈Λl;
2ξ → Ax, ξis continuous inξ∈Λlfor almost everyx∈Ω;
3Ax, tξ tp−1Ax, ξfor everyt≥0;
4KAx, 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 everyx ∈Ωand allξ, ζ∈Λl, whereK ≥ 1 is a constant. It should be noted that
Ax,∗:Ω×Λl → Λlis invertible and its inverse denoted byA−1satisfies similar conditions asAbut with H ¨older conjugate exponentqin place ofp.
Definition 1.2. If1.9is ap-harmonic type system, then we say the equation
d∗Ax, adu d∗b 1.10
is ap-harmonic type equation.
The following definition appears in9.
Definition 1.3. A differential formuis a weak solution for1.10inΩifusatisfies
Ω
Ax, adu, dϕd∗b, ϕ≡0 1.11
for everyϕ∈Wk,pΩ,Λl−1with compact support.
We can find that if we leta0 andb0, then thep-harmonic type system
Ax, adu bd∗v 1.12
becomes
Ax, du d∗v. 1.13
It is the conjugateA-harmonic equation, where the mappingA : Ω×Λl → Λlsatisfies the
following conditions:
If we letAx, ξ |ξ|p−2ξ, then the conjugateA-harmonic equation becomes the form
|du|p−2dud∗v. 1.15
It is the conjugatep-harmonic equation.
So we can see that the conjugatep-harmonic equation and the conjugateA-harmonic equation are the specificp-harmonic type system.
Remark 1.4. It should be noted that the mappingAx,∗inp-harmonic systemAx, adu bd∗v, is invertible. If we denote its inverse byA−1x,∗, then the mappingA−1x,∗:Λl →
Λlsatisfies similar conditions asAbut with H ¨older conjugate exponentqin place ofp.
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 in9, Section 4for the details.
Lemma 2.1. LetDbe a bounded, convex domain inRn. To eachy ∈ Dthere corresponds a linear
operatorKy:C∞D,Λl → C∞D,Λl−1defined by
Kyω
x;ξ1, . . . , ξl−1
1
0
tl−1ωtxy−ty;x−y, ξ1, . . . , ξl−1dt, 2.1
and the decomposition
ωdKyω
Kydω 2.2
holds at any pointy∈D.
We construct a homotopy operatorT :C∞D,Λl → C∞D,Λl−1by averagingK
y over all
pointsy∈D:
Tω
DϕyKyω dy, 2.3
whereϕformC∞Dis normalized so thatϕydy1.It is obvious thatωdKyω Kydω
remains valid for the operatorT :
ωdTω Tdω. 2.4
We define thel-formsωD∈DD,ΛlbyωD|D|−1Dωydyforl0andωDdTωfor
l1,2, . . . , n,and allω∈W1,pD,Λl,1< p <∞.
The following definition can be found in [9, page 34].
Definition 2.2. Forω∈DD,Λl, the vector valued differential form
∇ω
∂ω ∂x1
, . . . , ∂ω
∂xn
consists of differential forms∂ω/∂xi ∈DD,Λl, where the partial differentiation is applied
to coefficients ofω.
The proof of9, Proposition 4.1implies the following inequality.
Lemma 2.3. For anyω∈LpD,Λl, it holds that
∇Tωp,D≤Cn, pωp,D 2.6
for any ball or cubeD∈Rn.
The following Poincar´e inequality can be found in [2].
Lemma 2.4. Ifu∈W01,pΩ, then there is a constantCCn, p>0such that
1 |B|
B
|u|pχdx
1/pχ
≤Cr
1 |B|
B
|∇u|pdx
1/p
, 2.7
wheneverBBx0, ris a ball inRn, wheren≥2andχ2forp≥n,χnp/n−pforp < n.
Theorem 2.5. Letu∈DD,Λl, anddu∈LpD,Λl1. Then,u−u
Dis inLχpD,Λland
1 |D|
D|u−uD|
pχdx 1/pχ≤Cn, p, ldiamD
1 |D|
D|du|
pdx 1/p 2.8
for any ball or cubeD∈Rn, whereχ2forp≥nandχnp/n−pfor1< p < n.
Proof. We knowTdu u−uD. Now we supposeu−uQ Tdu IuIdxI, whereI
i1, . . . , il1take over alll1-tuples. So we have
∇Tdu
∂u ∂x1
, . . . , ∂u ∂xn
I
∂uI
∂x1
dxI, . . . ,
I
∂uI
∂xn
dxI
. 2.9
So we have
1 |D|
D
u−uDpχdx
1/pχ
1 |D|
D
I
uIdxI
pχ
dx
1/pχ
1 |D|
D
I
uI2
pχ/2
dx
1/pχ
.
By the inequality
n
i1
ai
2
1/2
≤n
i1
ai≤n1/2
n
i1
ai
2
1/2
2.11
for anyai≥0, and the Minkowski inequality, we have
1 |D|
D
I
uI2
pχ/2 dx 1/pχ ≤ I 1 |D|
D
uIpχdx 1/pχ
. 2.12
According to the Poincar´e inequality, we have
I
1 |D|
D|uI|
pχdx 1/pχ≤C
1n, pdiamD
I
1 |D|
D|∇uI|
pdx 1/p. 2.13
Combining2.10,2.12, and2.13, we can obtain
1 |D|
D
u−uDpχ
dx
1/pχ
≤C1n, pdiamD
I
1 |D|
D|∇uI|
pdx 1/p. 2.14
By2.9we have
∇Tdup,D
∂u ∂x1
, . . . , ∂u ∂xn p,D
D
∂x∂u1, . . . , ∂u ∂xn p dx 1/p D n
i1
∂x∂ui2
p/2 dx 1/p D n
i1
I ∂uI ∂xi 2 p/2 dx 1/p D I n
i1
Combining2.11and2.15, then we have
∇Tdup,D D I n
i1
∂uI ∂xi 2 p/2 dx 1/p
≥Cnl1
−1/2 D I n
i1
∂uI
∂xi
2
1/2p
dx
v
1/p
≥Cnl1
−1/2
D
I
n
i1
∂uI ∂xi 2 p/2 dx 1/p
≥Cnl1
−1/2
Cnl1
−p−1/p
I
D
n
i1
∂uI ∂xi 2 p/2 dx 1/p
≥C2n, p, l
−1
I
D ∇uIp
dx
1/p
,
2.16
whereC2n, p, l Cnl11/2p−1/p. Now combining2.14,2.16, and2.6, we can get
1 |D
D
u−uD|pχdx
1/pχ
≤C1n, pdiamD
I
1 |D|
D
∇uIpdx 1/p
≤C1n, p, lC2n, p, l
1 |D|
1/p
∇Tdup,D
≤C3n, p, ldiamD
1 |D|
D|du|
pdx 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 thep-harmonic type system. Before we prove the reverse H ¨older inequality, let us first see some lemmas.
Lemma 3.1. Iff, g≥0and for any nonnegativeη∈C∞0 Ω, it holds
Ωηf dx≤
Ωg dx, 3.1
then for anyh≥0:
Ωηfh dx≤
Proof. Letμbe a measure inX,fbe a nonnegativeμ-measurable function in a measure space X, using the standard representation theorem, we have
X
fqdμq
∞
0
tq−1μx:fx> tdt 3.3
for any 0< t < q.Now, we letμE Eηf dxandνE Eg dxthen, we can obtain
Ωηfh dx ∞
0
h>t
ηf dx dt≤
∞
0
h>t
g dx dt
Ωgh dx. 3.4
SoLemma 3.1is proved.
Lemma 3.2. Ifu, vis a pair of solution to thep-harmonic type system1.9, then it holds
Ω|η da|
pdx≤C
Ω|adudη|
pdx 3.5
for any nonnegativeη∈C0∞Ωand whereC Cl1 n p.
Proof. Sinceu, vis a pair of solutions toAx, adu bd∗v, it is also the solution to A−1x, bd∗v adu, whereA−1x,∗is the inverseAx,∗. Now, we suppose thatda
IωIdxI and letϕ1−IηsignωIdxI. By usingϕϕ1anddϕ1IsignωIdη∧dxI in
1.11, we can obtain
Ω
A−1x, bd∗v, dϕ1
da, ϕ1
dx≡0. 3.6
That is,
Ω
da,
I
ηsignωI
dxI
dx
Ω
A−1x, bd∗v,−
I
signωI
dη∧dxI
dx. 3.7
In other words,
Ω
I
ηωIdx
Ω
A−1x, bd∗v,−
I
signωI
dη∧dxI
dx. 3.8
By the elementary inequality
n
i1
ai2
1/2
≤n
i1
we have
Ωη|da|dx
Ωη
I
ωI2
1/2
dx≤
Ω
I
η|ωI|dx
Ω
A−1x, bd∗v,−
I
signωI
dη∧dxI
dx.
3.10
Using the inequality
|a, b| ≤ |a||b|, 3.11
3.10becomes
Ωη|da|dx≤
Ω
A−1x, bd∗v
I
signωI
dη∧dxI
≤
Ω
A−1x, bd∗v I
sign
ωI|dη|dx
Cnl1
Ω
A−1x, bd∗v|dη|dx
Cnl1
Ω|adu||dη|dx,
3.12
whereItakes over alll1-tuples fordη∈Λl1, thus it hasCl1
n numbers at most. Now we
letf|da|andg Cl1
n |adu||dη|. In the subset{x:fηg}, we have
{x:fηg}|ηda| pdx≤
{x:fηg}|adudη|
pdx. 3.13
In the subset{x:fη /g}, leth |fη|p− |g|p/fη−g, then we easily obtainh >0. So by
Lemma 3.1, we have
{x:fη /g}hfη dx≤
{x:fη /g}hg dx. 3.14
That is to say
that is,
{x:fη /g}|fη| pdx≤
fη /g
|g|pdx. 3.16
Combining3.13and3.16, we have
Ω|fη| pdx≤
Ω|g|
pdx, 3.17
that is,
Ω|η da| pdx≤
Ω
Cln1adudηpdx. 3.18
SoLemma 3.2is proved.
The following lemma appears in2.
Lemma 3.3. Suppose that0< q < p < s≤ ∞,ξ∈R, and thatBBx0, ris a ball. If a nonnegative
functionv∈LpB, dμsatisfies
1 μλB
λB
vsdμ
1/s
≤C1−λξ
1 μB
B
vpdμ
1/p
3.19
for each ballBBx0, rwithr≤rand for all0< λ <1, then
1 μλB
λB
vsdμ
1/s
≤C1−λξ/θ
1 μB
B
vqdμ
1/q
∀0< λ <1. 3.20
HereC >0is a constant depending onp, q, sandθ∈0,1is such that1/pθ/q 1−θ/s.
The following lemma appears in10.
Lemma 3.4. Letu, vbe a pair of solutions of thep-harmonic type system on domainΩ, then we have a constantConly depending onK, n, p, andl, such that
wherecis any closed form (i.e.,dc 0) and for anyη ∈ C∞0Ω. Also we have a constantConly depending onK, n, q, such that
η d∗vq,Ω≤Cv−cdηq,Ωηbq,Ω, 3.22
wherecis any coclosed form (i.e.,d∗c0) andqis the conjugate exponent ofp.
Theorem 3.5. Ifu, vis a pair of solutions to thep-harmonic type system, then there exists a constant
C >0dependent onK, p, n,andl, such that
1 |Q|
Q
u−uQa∞,Qsdx 1/s
≤C1−σ−1−tχ/pχ−1diamQ1χ/χ−1
×
1 |σQ|
σQ
u−uσQa∞,σQtdx
1/t 3.23
for any0< s, t <∞,σ >1and all cubes withQ⊂σQ⊂Ω, whereχ >1is the Poincar´e constant.
Proof. Suppose that the center ofQisx0and diamQr, 0< λσ−1 <1. Let
rmλ 1−λ2−m, m0,1,2, . . . . 3.24
Thenrm is decreasing and λ < rm < 1. So we haveuQ|rmQ urmQ, for anym ∈ 0,1,2, . . . .
Let ηm ∈ C0∞rmQ be a nonnegative function such thatηm 1 inrm1Q, 0 ≤ ηm ≤ 1 in
rmQ−rm1Q.|dηm| ≤1−λ−12mr−1. Given anyt≥0 and letωm |u−uQ|a∞,Q1t/pηm,
then we have
dum
1 t
p u−uQa∞,Q
t/p
ηmdu−uQu−uQa∞,Q1t/pdηm. 3.25
By the Minkowski inequality, we can obtain
rmQ
dumpdx 1/p
≤
rmQ
u−uQa∞,Qptdηmpdx 1/p
pt
p
rmQ
du−uQpu−uQa∞,Qtηmpdx 1/p
.
3.26
We assume thatu−uQ IaIdxI, then we have|u−uQ| Ia2I
1/2. Ifu−u
Qis zero, then
that|∇Tdu| Ini1|∂aI/∂xi|21/2
du−uQ∇u−uQ
∂u−uQ
∂x1
, . . . ,∂u−uQ ∂xn
n
i1
∂u−uQ
∂xi 2 1/2 n
i1
∂u−uQ
∂xi 2 1/2 n
i1
∂Ia2I1/2 ∂xi
21/2
n
i1
1
Ia2I
I aI ∂aI ∂xi
21/2
≤
n
i1
1
Ia2I
I
a2I
I
∂aI
∂xi
21/2
n
i1
I
∂aI
∂xi
21/2
n
i1
I ∂aI ∂xi 2 1/2
∇Tdu∇u−uQ.
3.27
So we have
du−uQ≤∇Tdu. 3.28
For anyη∈C∞0Ω, according to2.6, we have
η∇T dωp,D≤Cn, pmax
x∈Dηdωp,D. 3.29
By the similar method asLemma 3.1, we can prove the following inequality:
rmQ
du−uQp
u−uQa∞,Qtηmpdx 1/p
≤
rmQ
ηmp∇Tdupu−uQa∞,Q
t
dx
1/p
≤Cn, pmax
x∈D
ηpm
rmQ
ηmp|du|pu−uQa∞,Qtdx 1/p
for anyη∈C0∞Ω. ByLemma 3.1and3.21, we can obtain
rmQ
ηmp|du|pu−uQa∞,Q
t
dx
1/p
≤2C
rmQ
ηmp|a|pu−uQa∞,Qtdx 1/p
2C
rmQ
dηmp
u−uQa∞,Qptdx 1/p
≤2C
rmQ
ηmpap∞,Qu−uQa∞,Q
t
dx
1/p
2C
rmQ
dηmp
u−uQa∞,Qptdx 1/p
≤2C
rmQ
ηmpu−uQa∞,Q
pt
dx
1/p
2C
rmQ
dηmpu−uQa∞,Qptdx 1/p
.
3.31
Combining3.26,3.30, and3.31, by the values ofηm, we have
rmQ dump
dx
1/p
≤C1pt
1 1−λ−12mr−1
rmQ
u−uQa∞,Qpt
dx
1/p
.
3.32
Forηm 1 inrm1Qand 0 ≤ ηm ≤ 1 inrmQ−rm1Q, and as we have |rm|/rm1 |λ 1−
λ2−m|/λ 1−λ2−m−1≤2, so we have|rmQ|/|rm1Q| ≤2n. By the Poincar´e inequality, we
know
1 |rm1Q|
rm1Q
u−uQa∞,Qχptdx 1/pχ
≤ 1 rm1Q
rmQ
ηpχmu−uQa∞,Qχptdx1/pχ
≤
1
rm1Q
rmQ
umpχdx 1/pχ
≤2n
1
rmQ
rmQ umpχ
dx
1/pχ
≤C2rmr
1
rmQ
rmQ
dumpdx 1/p
≤C3rmrpt
1 1−λ−12mr−1
rmQ
u−uQa∞,Qptdx 1/p
≤C3pt1−λ−12m1r
rmQ
u−uQa∞,Qptdx 1/p
.
Now we setκpt, then by computation, we obtain
1
rm1Q
rm1Q
u−uQa∞,Qκχdx 1/κχ
≤C3
p/κ
κp/κ1−λ−p/κ2pm/κr1p/κ
×
1
rmQ
rmQ
u−uQa∞,Qκ
dx
1/κ
.
3.34
Since this inequality holds for allκ > p, it can be applied with κ κm pχm. And we
can easily prove1/|Q|Q|f|pdx1/p is increasing withpand its limit is ess supQ|f|. So by iterating we arrive at the desired inequality forqp:
ess sup
λQ
u−uQa∞,Q
≤ lim
m→ ∞
1
rmQ
rmQ
u−uQa∞,Qκmχ
dx
1/κmχ
≤C4
1−λ−11rΣ∞i0χ−m∞
m0
2mχ−m∞ m0
pχmχ−m
×
1 |Q|
Q
u−uQa∞,Qpdx 1/p
≤C51−λ−χ/χ−1r1χ/χ−1
1 |Q|
Q
u−uQa∞,Qp
dx
1/p
.
3.35
We can observe that the constantsC5andχare independent ofx0andrin3.35, thus
3.35holds not only in the cubeQQx0, rbut also in each ball insideQ. By Lemma3.5
we can obtain
1 |λQ|
λQ
u−uQa∞,Qsdx 1/s
≤C51−λ−θχ/χ−1r1χ/χ−1
×
1 |Q|
Q
|u−uQ|a∞,Qtdx
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≤ ∞. Since1/|Q|Q|f|pdx1/pis increasing withp,
1 |λQ|
λQ
u−uQa∞,Qsdx 1/s
≤C51−λ−tχ/pχ−1r1χ/χ−1
×
1 |Q|
Q
|u−uQ|a∞,σQtdx
for any 0< s <∞and 1< p < t <∞. Combining3.36and3.37, we have
1 |Q|
Q
u−uQa∞,Qsdx 1/s
≤C61−λ−tχ/pχ−1r1χ/χ−1
×
1 |σQ|
σQ
|u−uσQ|a∞,σQtdx
1/t 3.38
for any 0< s, t <∞andσ >1 such thatσQ⊂Ω.Theorem 3.5is proved.
Acknowledgment
This work is supported by the NSF of ChinaGrants nos. 10671046 and 10771044.
References
1 S. Ding, “Some examples of conjugatep-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 forA-harmonic tensors,”Illinois Journal of Mathematics, vol. 43, no. 4, pp. 613–632, 1999.
4 G. Bao, “Arλ-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 forA-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 theA-harmonic equation,”Journal of
Mathematical Analysis and Applications, vol. 279, no. 1, pp. 350–363, 2003.
7 L. D’Onofrio and T. Iwaniec, “Thep-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 nonhomogeneousA -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.
