\(L^{\varphi}\) Embedding inequalities for some operators on differential forms

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R E S E A R C H

Open Access

L

ϕ

-Embedding inequalities for some

operators on differential forms

Guannan Shi

1

_{, Shusen Ding}

2

_{and Yuming Xing}

1*

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Harbin Institute of Technology, Harbin, 150001, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, the local Poincaré inequality and embedding inequality are proved ﬁrst. Then the global embedding inequality of composite operators for diﬀerential forms onLϕ_{-averaging domains with}_Lϕ_{-norm is established. Some examples are also given}

to illustrate applications.

Keywords: homotopy operator; Dirac operator; Green’s operator;Lϕ-averaging domains

1 Introduction

The purpose of this paper is to establish the global Sobolev embedding inequality with Lϕ_{-norm for the composite operators applied to diﬀerential forms on}_Lϕ_-averaging

do-mains. As extensions of functions, differential forms have been deeply studied and widely applied in many fields, such as global analysis, nonlinear analysis, PDEs and differential geometry; see [–] for more information. The Sobolev embedding inequality, as a fun-damental tool in the Sobolev space of functions, has been very well studied in []. The homotopy operatorT, Dirac operatorD, and Green’s operatorGare three key operators acting on differential forms and each of them has received much attention recently. How-ever, the compositions of these operators are so complicated that the results concerned on them are quite few. Even though they are very complicated, the study for the composite operators has been increasing during the recent years; see [–, ]. In many cases, we need to study or to use the compositions of operators. For example, we have to deal with the composition of the Dirac operatorDand Green’s operatorGwhen we consider the Poisson equationD_{(G(u)) =}_u_–_{H(u), where}_D₌_and_H_{is the projection operator. In}

, Ding and Liu initiated the study of the compositionD◦Gand obtained some basic inequalities in []. During our recent investigation of the operator theory of diﬀerential forms, we realized that for applying the decomposition theorem toD◦G, it is necessary to deal with the compositionT◦D◦G. Hence, we are motivated to study the composition

T◦D◦G. We obtain some basic estimates forT◦D◦Gand establish the global Sobolev

embedding inequality withLϕ_-norm

TDG(u) –TDG(u)_E_W,ϕ

E ≤CuL ,ϕ

E ,

whereEis anLϕ_{-averaging domain [].}

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For convenience, we keep using the traditional notations and terminologies. Except for special instructions, E⊆ Rn _{is a bounded domain, and} _|_E_| _{denotes the Lebesgue} measure of E, n≥. Suppose thatBr

x is a ball with a radiusr, centered at x. For any

σ > ,B⊆E andσB⊆Ehave the same center and satisfy diam (σB) =σdiam(B). Let

l(Rn_{) be the space of all}_{l-forms in}_Rn_{, which is expanded by the exterior product of} eB₌_ei∧_ei∧ · · · ∧_eil_{, where}_B_{= (i}

, . . . ,il), ≤i<· · ·<il≤n,l= , , . . . ,n.C∞(lE) is the space of a smoothl-form onE. We useD(E,l) to denote the space of all diﬀerential l-forms onE, that is,u(x) belongs toD(E,l) if and only if there exist somelth-diﬀerential functionsuBinEsuch thatu(x) =

BuB(x)dxB=

uii···il(x)dxi∧dxi∧ · · · ∧dxil.

Lp_(E,l_{) is a Banach space with the norm equipped by}_u_p

,E= (

E|u(x)|pdx)/p, where u(x)∈D(E,l_{) and every coeﬃcient function}_u

B∈Lp(E),  <p<∞. In fact,u(x) onE

is the Schwarz distribution. Ifw(x) >  a.e. andw(x)∈L

loc(Rm),w(x) is called a weight.

Letdμ=w(x)dx, thenLp_(E,l_,_{w) is a weighted Banach space with the norm expressed}

byu(x)p,E,w= (

E|u(x)|pw(x)dx)/p. In this notation, the exterior derivative is denoted by dand the Hodge codiﬀerential operator is expressed byd_{. Refer to [] for more details.}

Moreover, the Dirac operator, designated byD, was initially set forth by Paul Dirac, and we get its extensions, such as the Hodge-Dirac operator and the Euclidean Dirac opera-tor. In this paper, the Dirac operator we choose is the Hodge-Dirac operatorD=d∗+d. The Green’s operator is a bound self-adjoint linear operator. It is commonly used to de-fine the Poisson equation for differential forms; here it is represented byG. The homotopy operatorT[], advanced by Iwaniec, is defined onC∞(D,l_{) and constructed as follows:}

Tu=

D

ψ(y)Kyu dy,

where the linear operatorKyis denoted by (Kyu)(x;ξ, . . . ,ξl–) =



tl–u(tx+y–ty;x–

y,ξ, . . . ,ξl–)dt, andψ fromC∞(D) is normalized so that

ψ(y)dy= .

From [], we obtain some good results. For any diﬀerential formu, the decomposition below holds:

u=d(Tu) +T(du). (.)

Meanwhile, foru ∈D( ,l), we deﬁne

u =

| |– _u(y)_dy, _l_{= ,}

dT(u), l= , , . . . ,n. (.)

Then we know that, for any diﬀerential formu∈Ls

loc(B,l),l= , . . . ,n,  <s<∞,

_∇(Tu)_s_,_B≤C|B|us,B and Tus,B≤C|B|diam(B)us,B (.)

holds for any ballBin .

Concerning the homotopy operatorT, according to [], we can obtain a nice property. That is, ⊂Rn_{is the union of a collection of cube}

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() = ∞_k_= k;

() 

i ∩ j =∅, ifj=i;

() there exist two constantsC> andC> , such that

Cdiam( k)≤distance( k,F)≤Cdiam( k).

Based on this fact, the homotopy operatorTcan be extended to any domain ⊂Rn. For anyx∈ ,x∈ kfor somek, letT k be the homotopy operator deﬁned on k, which is

the convex and bounded set. So, the following deﬁnition forT on any domain holds:

T =

+∞

k=

T kχ k,

whereχ _kis the characteristic function deﬁned on .

Further, it is essential to recall some well-known results as regards theA-harmonic equa-tion for diﬀerential forms, which appeared in []. To be more precise, we here consider the non-homogeneousA-harmonic equation as follows:

dA(x,du) =B(x,du), (.)

whereA:E× ∧l₍_Rn₎_{→ ∧}l₍_Rn_{) and}_B_:_E_{× ∧}l₍_Rn₎_{→ ∧}l–₍_Rn_{) satisfy the conditions}

_A(x,ξ)≤a|ξ|p–, A(x,ξ)·ξ≥ |ξ|p and B(x,ξ)≤b|ξ|p–

for almost everyx∈Eand allξ∈ ∧l(Rn_{). Meanwhile, the parameters}_a,_b_{>  and  <}_p_< ∞associated with (.) are constants and a ﬁxed exponential, respectively. IfB= , the equationd_A(x,_{du) =  is called a homogeneous}_{A-harmonic equation.}

In addition, for proving the global embedding inequality on theLϕ_{-averaging domain,}

we also need the following deﬁnitions and notations.

A function ϕ(x) is called an Orlicz function, ifϕ(x) satisﬁes: ()ϕ(x) is continuously increasing; ()ϕ() = . Furthermore, if the Orlicz functionϕ(x) is a convex function,ϕ(x) is named a Young function. Therefore, based on this type of functions, the Orlicz norm for diﬀerential forms can be denoted as follows.

Take ϕ(x) as a Orlicz function, and E⊂ Rn as a bounded domain, for any u(x)∈ Lp_loc(E,l_),_l_{= , , , . . . ,}_{n, the Orlicz norm for the diﬀerential form is equipped with}

u_Lϕ

E=inf

λ>  :

E

ϕ _|

u|

λ

dμ≤

,

where the measureμis expressed bydμ=w(x)dx,w(x) is a weight.

It is easy to prove thatLϕ_E is a Banach space. Actually, provided thatϕ(x) is taken as

ϕ(x) =xs(s> ), thenϕ(x) is an Orlicz function, and it trivially corresponds to aLs(μ) space. As a result of that, we can say thatLϕ_Eis the generalization ofLs_E.

Similarly, we intend to giveLϕ_{-averaging domains. Based on the Orlicz norm above, the}

Orlicz-Sobolev space ofl-form is denoted byW,ϕ_(E,l_{), with the norm} u_W,ϕ

E =uW

,ϕ₍_E_,l₎=diam(E)–u_Lϕ

E+∇uL

ϕ

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From the deﬁnition ofW_E,ϕ, we know thatW_E,ϕ is equal toLϕ_E∩Lϕ_,_E. In detail, ifϕ(x) is expressed byϕ(x) =ts(s> ), we can conclude the norm ofW_E,sfor the diﬀerential form:

u_W,s

E =uW,s(E,l)=diam(E)

–_u_L_s

E+∇uLsE.

Besides that, we will show another deﬁnition, which was initially introduced by Ding.

Deﬁnition . [] Letϕ(x) be a Young function, the proper domainE⊆Rnis called

theLϕ_{-averaging domain, if}_μ_{(E) <}_∞_{and there exists a constant}_C_{>  such that for any}

B⊆Eandϕ(|u|)∈Lloc(E,μ),usatisﬁes



μ(E)

E

ϕτ|u–uB|

dμ≤Csup

B⊂E 

μ(B)

B

ϕσ|u–uB|dμ,

where the measureμis denoted bydμ=w(x)dx,w(x) is a weight,σandτ are constants with  <τ,σ< , and the supremum is over all ballsB⊂Ewith B⊂E.

Using the same analysis method as of theLϕ_E-norm, we can conclude thatLϕ_-averaging

domains are the generalization ofLs_{-averaging domains.}

2 Main results

Before the main results given, we will make some restrictions to the Young functionϕ(x). Here, we let the Young functionϕ(x) belong to theG(p,q,C)-class (≤p<q<∞,C≥), that is, for anyt> , the Young functionϕ(x) satisﬁes:

() /C≤ϕ(t/p_)/f_(t)_≤_C_;

() /C≤ϕ(t/q)/g(t)≤C,

wheref andgare increasingly convex and concave functions deﬁned on [,∞], respec-tively.

Now, we establish four important theorems based on the above-mentioned conditions.

Theorem . Let T be the homotopy operator, D be the Dirac operator, and G be the

Green’s operator.Meanwhile,we assume thatϕ(|u|)∈L_loc(E),u∈C∞(lE)is a solution of

the non-homogeneous A-harmonic equation,the Young function imposed a doubling

prop-ertyϕ(x)belongs to the G(p,q,C)-class,and the bounded subset E⊆Rn_{is the L}ϕ_-averaging

domain.Then,for any ball B⊆E,we get

TDG(u) –TDG(u)_B_Lϕ

B≤C

diam(B)u_Lϕ σB,

whereσB⊆E andσ> is a constant.

Theorem . Let T be the homotopy operator, D be the Dirac operator,and G be the

Green’s operator.Meanwhile,we assume thatϕ(|u|)∈L

loc(E),u∈C∞(lE),is a solution

of the non-homogeneous A-harmonic equation,the Young functionϕ(x)having imposed

a doubling property belongs to the G(p,q,C)-class and the bounded subset E⊆Rnis the

Lϕ_{-averaging domain.}_Then,_{for any ball B}_⊆_E,_{we get}

TDG(u) –TDG(u)_B_W,ϕ

B ≤C

diam(B)u_L,ϕ σB,

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Based on the above theorem, we cannot only establish the following global embedding inequality onLϕ_{-averaging domains, but we also get the global Poincaré-type inequality}

fortunately.

Theorem . Let T be the homotopy operator, D be the Dirac operator,and G be the

Green’s operator.Additionally,we assume thatϕ(|u|)∈L

a doubling property belongs to the G(p,q,C)-class and the bounded subset E⊆Rn_{is the}

Lϕ_{-averaging domain.}_Then,_{for any ball B}_⊆_E,_{we have}

E ≤CuL ,ϕ

E .

Theorem . Let T be the homotopy operator, D be the Dirac operator,and G be the

Green’s operator.Additionally,we assume thatϕ(|u|)∈L

a doubling property belongs to the G(p,q,C)-class and the bounded subset E⊆Rn_{is the}

Lϕ_{-averaging domain.}_Then,_{for any ball B}_⊆_E,_{we have}

TDG(u) –TDG(u)_B

LϕE≤CuL

ϕ

E,

where B⊆B is a ﬁxed ball.

3 Preliminary results

For proving the theorems in Section , we shall show and demonstrate some lemmas in this section.

Lemma .[] Let <p,q<∞,and _t =_p +_q,if f and g are the measurable functions

deﬁned onRn_,_then

fgt,I≤ fp,I· gq,I

for any I⊆Rn_.

The inequality in Lemma . is actually the generalized Hölder inequality. Speciﬁcally, ift=  and  <p,q<∞, the inequality above is the classical Hölder inequality.

Lemma . [] Let u be a solution of the non-homogeneous A-harmonic equation in

a domain , and  <s,t<∞, then there exists a constant C, independent of u, such

that

us,B≤C|B|

t–s st _ut_,_σ_B

for all balls B withσB⊂E,whereσ> is some constant.

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Lemma .[] Suppose that the diﬀerential form u∈Lp_(E,l_),_l_{= , . . . ,}_n,_{and G is the}

Green’s operator,then there exists a constant C,independent of u,such that

dd∗G(u) +d∗dG(u) +d+d∗G(u) +G(u)_p_,_B≤Cup,B

for any B⊂ .

Because=d∗d+dd∗andD=d∗+d, by using Lemma ., it implies that

_G(u)

p,B≤Cup,B and DG(u)p,B≤Cup,B.

Lemma .[] If u∈Lp(D,l),  <p<∞,then uD∈Lp(D,l)and

uDp,D≤Ap(n)μ(D)uD.

According to the above results, we can prove a very useful lemma as follows.

Lemma . Let T be the homotopy operator,D be the Dirac operator,and G be the Green’s

operator.Additionally,we assume that u∈C∞(lE)is a solution of the non-homogeneous

A-harmonic equation, the Young function ϕ(x)belongs to the G(p,q,C)-class, and the

bounded subset E⊆Rn_{is the L}ϕ_{-averaging domain.}_Then,_{for any ball B}_⊆_E,_{we have}

TDG(u) –TDG(u)_B_s_,_B≤C|B|diam(B)us,B

for any B⊂E,where s> .

Proof Notice thatTDG(u) is at least diﬀerential -form, therefore, by using (.) and

re-placingubyTDG(u), we have

TDG(u) =TdTDG(u)+dTTDG(u).

It follows from (.) thatdT(TDG(u)) = (TDG(u))B, that is,

TDG(u) –TDG(u)_B_s_,_B=TdTDG(u)_s_,_B.

Applying (.), Lemma ., and Lemma ., we obtain

TDG(u) –TDG(u)_B_s_,_B =TdTDG(u)_s_,_B

≤C|B|diam(B)d

TDG(u)_s_,_B ≤C|B|diam(B)DG(u)

Bs,B ≤C|B|diam(B)DG(u)_s_,_B

≤C|B|diam(B)us,B.

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Note By observing the proving process, we can get a valuable result as follows:

TdTDG(u)_s_,_B≤C|B|diam(B)us,B.

Lemma . Let T be the homotopy operator,D be the Dirac operator,and G be the Green’s

operator.Additionally,we assume that u∈C∞(l_E)_{is a solution of the non-homogeneous}

A-harmonic equation,the Young functionϕ(x)having imposed a doubling property belongs

to the G(p,q,C)-class and the bounded subset E⊆Rn_{is the L}ϕ_{-averaging domain.}_Then,

for any ball B⊆E,we have

TDG(u)_Lϕ

B≤Cdiam(B)uL

ϕ σB,

where the constantσ> andσB⊆E.

Proof From (.), we have

TDG(u)_s_,_B≤Cdiam(B)|B|DG(u)_s_,_B

≤Cdiam(B)|B|us,B. (.)

Becauseϕbelongs to theG(p,q,C)-class, we have

B

ϕTDG(u)dx

=g·g–

B

ϕTDG(u)dx

≤g

B

g–ϕTDG(u)dx

≤g

B

CTDG(u)

q dx

≤Cϕ

B

CTDG(u)

q dx

 q

.

Sinceϕis doubling, according to (.) and Lemma ., we have

B

ϕTDG(u)dx

≤Cϕ

C|B|+

 n_uq

,B

≤Cϕ

|B|+n|_B|  p–q

σB |u|pdx

 p

≤Cf

|B|p(+n)|_B| p q–

σB |u|pdx

≤C

σB

f|B|p–+p(q+n)₊|_u|p_dx

≤C

σB Cϕ

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≤C|B|+

 q+n–p

σB

ϕ|u|dx

≤C|B|

 n

σB

ϕ|u|dx. (.)

Therefore, by using|B|n ₌_C__diam_{(B), we can get the following result:}

TDG(u)_Lϕ

B≤Cdiam(B)uL

ϕ σB,

whereσ> .

This is the end of the proof of Lemma ..

Lemma . Let T be the homotopy operator,D be the Dirac operator,and G be the Green’s

operator.Additionally,we assume thatϕ(|u|)∈L_loc(E)and u∈C∞(lE)is a solution of

the non-homogeneous A-harmonic equation,the Young functionϕ(x)having imposed a

doubling property belongs to the G(p,q,C)-class and E is a bounded domain.Then we get

_∇TdTDG(u)_Lϕ

B≤C|B|

diam(B)u_Lϕ σB

for all balls B withσB⊂E,whereσ> is some constant.

Proof According to the elementary inequality (.) of the homotopy operatorTand

Lem-ma ., we have

_∇TdTDG(u)_p_,_B≤C|B|d

TDG(u)_p_,_B≤C|B|DG(u)_p_,_B. (.)

Applying (.) and Lemma ., we get

_∇TdTDG(u)_p_,_B≤C|B|diam(B)up,B.

Using the same method as used in Lemma ., the following inequality holds:

_∇_Td

TDG(u)_Lϕ

B≤Cdiam(B)uL

ϕ σB.

The proof of Lemma . is completed.

Lemma .(Covering lemma) [] Each domain has a modiﬁed Whitney cover of cubes

V={Qi}such that

i

Qi= ,

Qi∈V χ_

Qi≤N χ

and some N > ,and if Qi∩Qj=∅,then there exists a cube R(this cube does not need

to be a member ofV)in Qi∩Qjsuch that Qi∪Qj⊂NR.Moreover,if isδ-John,then

there is a distinguished cube Q∈Vwhich can be connected with every cube Q∈V by a

chain of cubes Q,Q, . . . ,Qk=Q fromV and such that Q⊂ρQi,i= , , , . . . ,k,for some

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4 Demonstration of main results

According to the above deﬁnitions and lemmas, we will prove four theorems in detail. First of all, let us prove Theorem ..

Proof of Theorem. First of all, similar to the proof of Lemma ., we have

B=

TdTDG(u)_Lϕ

B. (.)

This means that the key point is to prove the following inequality:

TdTDG(u)_Lϕ

B≤CuL

ϕ σB.

According to the properties of theG(p,q,C)-class, we have

B

ϕTdTDG(u)dx

=g·g–

B

ϕTdTDG(u)dx

≤g

B

g–ϕTdTDG(u)dx

≤g

B CTd

TDG(u)qdx

≤Cϕ

B CTd

TDG(u)qdx

 q

.

Becauseϕis doubling, using Lemma ., we have

B

ϕTdTDG(u)dx

≤Cϕ

C|B|+

 n_uq_,_B

≤Cϕ

|B|+n|_B|  p–q

σB |u|pdx

 p

≤Cf

|B|p(+n)|_B| p q–

σB |u|pdx

≤C

σB

f|B|p–+p(q+n)|_u|p_dx

≤C

σB Cϕ

|B|+q+n–p|_u|_dx

≤C|B|+

 q+n–p

σB

ϕ|u|dx

≤C|B|

 n

σB

ϕ|u|dx. (.)

Combining (.) with (.) and using|B|n₌_C__diam_{(B) yield}

B

ϕTDG(u) –TDG(u)_Bdx≤Cdiam(B)

σB

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As a result, we obtain

B

ϕ _|

TDG(u) – (TDG(u))B|

λ

dx≤Cdiam(B)

σB

ϕ _|

u|

λ

dx

for anyBwithσB⊂Eand any constantλ> . It means that the following inequality on Lϕ_E-averaging domains holds:

B≤Cdiam(B)uL

ϕ σB.

The proof of this theorem is ﬁnished.

Remark From the proof of Theorem ., it is easy to derive the following inequality:

TdTDG(u)_Lϕ

B≤Cdiam(B)uL

ϕ

σB. (.)

We shall prove Theorem . by using Theorem . and Lemma ..

Proof for Theorem. Using (.) repeatedly, we have

B =

TdTDG(u)_W,ϕ

B . (.)

Combining (.) with (.), and applying Theorem . and Lemma ., we have

B =diam(B)

–_Td_TDG(u)

Lϕ_B+∇Td

TDG(u)_Lϕ

B

≤diam(B)–Cdiam(B)uLϕ_σ

B

+CuLϕ_σ

B

≤CuLϕ_σ

B+CuL

ϕ σB

≤Cu_Lϕ σB,

whereσ=max{σ,σ}andσ> , for all ballsBwithσB⊂E.

This is the end of the proof of Theorem ..

Proof for Theorem. From the covering lemma, Lemma ., and Lemma ., we have

_∇TdTDG(u)_Lϕ

E≤

B∈V

_∇TdTDG(u)_Lϕ

B

≤

B∈V

C|B|uLϕ

σB

≤CNuLϕ_E

≤CuLϕ_E. (.)

Similarly, using the covering lemma, Lemma ., and (.) implies

TdTDG(u)_Lϕ

E≤

B∈V

TdTDG(u)_Lϕ

B

≤

B∈V

Cdiam(B)uLϕ_σ_B

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≤Cdiam(E)NuLϕ_E

≤Cdiam(E)uLϕ_E. (.)

Thus, from (.), (.), and (.), we obtain

E =

TdTDG(u)_W,ϕ

E

=diam(E)–TdTDG(u)_Lϕ

E+

_∇TdTDG(u)_Lϕ

E

≤diam(E)–Cdiam(E)uLϕ

E

+CuLϕ

E

≤CuLϕ

E.

We have completed the proof of Theorem ..

Next, we will prove Theorem . by using Deﬁnition . and Theorem ..

Proof of Theorem. According to Deﬁnition ., we have

|E|–

E

ϕTDG(u) –TDG(u)_B

dx≤ sup

B⊂E| B|–

B

ϕTDG(u) –TDG(u)_Bdx

≤ sup

B⊂E|

B|–Cdiam(B)

σB

ϕ|u|dx.

Becausesup_B_⊆_E_Eϕ(|u|)dxdoes not depend onB, we obtain

|E|–

E

ϕTDG(u) –TDG(u)_B

dx≤ sup

B⊂E|

B|–Cdiam(B)

E

ϕ|u|dx.

Therefore, we have

E

ϕλ–TDG(u) –TDG(u)_B

dx≤C

E

ϕλ–|u|dx.

We ﬁnish the proof of Theorem ..

In addition, we can obtain a global estimate about the composite operators using the same method as of Theorem ..

Corollary . Let T be the homotopy operator, D be the Dirac operator,and G be the

Green’s operator. Additionally, we assume that u∈C∞(lE) is a solution of the

non-homogeneous A-harmonic equation,the Young functionϕ(x)belongs to the G(p,q,C)-class,

and the bounded subset E⊆Rn_{is the L}ϕ_{-averaging domain.}_Then,_{for any ball B}_⊆_E,_we

have

TDG(u)_Lϕ

E≤Cdiam(B)uL

ϕ

E.

Remark If we chooseϕ(x) =xs_{, we have}

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5 Applications

In this section, we will discuss the applications of the results obtained.

Example . Letr>  and ={(x,x,x) :x+x+x≤r} ⊂R. Consider the -form

u(x,x,x) =

x

 + (x

+x+x)

dx+

x

 + (x

+x+x)

dx+

x

 + (x

+x+x)

dx

deﬁned in . Thenuis a solution of the non-homogeneousA-harmonic equation for any operatorsAandBsatisfying the required conditions.

As the application of the obtained theorems, our goal is to get the upper bound of TDG(u) satisfying the above conditions. Normally, we would consider calculating the inte-gral ofTDG(u), however, one will see thatTDG(u) with theLϕ_{-norm is very complicated.}

In this case, we can use Theorem . to get the estimate ofTDG(u) with theLϕ_-norm

inW,ϕ.

Initially, according to the condition and the expression ofuin Example ., we see that

u(x,x,x) 

= x

 

( + (x

+x+x))

+ x

 

( + (x

+x+x))

+ x

 

( + (x

+x+x))

.

Furthermore, becauseuis deﬁned in and (x

+x+x)> , we get

u(x,x,x)≤. (.)

As a result, by using Theorem . and (.), we have

TDG(u) –TDG(u) _W,ϕ≤C_uLϕ ≤C_Lϕ ≤C_r.

The above example can be extended to the case ofRn_{. Particularly, we can check that the} -form deﬁned inRn_,

u(x, . . . ,xn) = n

i=

xi  + (x

+· · ·+xn) dxi,

is a solution of the non-homogeneousA-harmonic equation for any operatorsAandB satisfying the required conditions. So, we can also apply Theorem . to this extended case as we did in Example ..

To end this section, we take a -form inRas an example.

Example . Letk,m>  be constants. Consider a diﬀerential -form inR_,

u(x,y,z) = k

m+x₊_y₊_zdx∧dy∧dz.

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Similarly, applying the same method, we can obtain the following result as regards

TDG(u) in any bounded set ⊂R_:

u(x,y,z)≤ k m

and

TDG(u) –TDG(u) _W,ϕ≤CuLϕ ≤Ck/mLϕ.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All results and investigations of this manuscript were due to the joint efforts of all authors. GS mainly studied the theoretical proofs and drafted the manuscript. SD and YX conceived of the initial idea of this manuscript and improved the final version. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China.}2_{Department of Mathematics,}

Seattle University, Seattle, WA 98122, USA.

Acknowledgements

The authors express their deep appreciation to the referees’ eﬀorts and suggestions, which highly improved the quality of this paper.

Received: 14 August 2015 Accepted: 6 November 2015

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