Embedding theorems for composition of homotopy and projection operators

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R E S E A R C H

Open Access

Embedding theorems for composition of

homotopy and projection operators

Jinling Niu

1

_{, Shusen Ding}

2

_{and Yuming Xing}

1*

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

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

Abstract

We prove both local and global embedding theorems withLϕ-norms for the composition of the homotopy operator and projection operator applied to diﬀerential forms. We also establish someLϕ_{-norm inequalities for certain}

compositions of the related operators.

MSC: Primary 35J60; secondary 35B45; 30C65; 47J05; 46E35

Keywords: embedding inequalities; diﬀerential forms; homotopy operators; projection operators

1 Introduction

The homotopy operatorT and projection operatorH deﬁned on diﬀerential forms are two critical operators which have been very well studied and used in recent years, see [–]. In many situations, we need to deal with the compositionT◦Hof the homotopy operatorTand the projection operatorH. For example, when we consider the decompo-sition ofH(u), we have to face the compositionT◦H. The study of the compositionT◦H

of homotopy and projection operators was initiated by Ding and Liu in  in [] and [], respectively, where they investigated singular integrals of this composite operator and established someLp_{inequalities for the composite operator}_T_◦_H_{with singular factors.}

Later, in , Bi and Ding proved someLϕ_{-estimates for this composite operator}_T_◦_H

in [], whereϕsatisﬁes theG(p,q,C) condition. The purpose of this paper is to establish theLϕ_{-embedding theorems for the composition}_T_◦_H_{applied to diﬀerential forms, here}

ϕsatisﬁes theNG(p,q) condition. If we chooseϕ(t) =tp_{, the}_Lϕ_{-norm inequalities reduce}

toLp_{-norm inequalities. Our main}_Lϕ_{-embedding inequality for the composite operator}

can be simply stated as

_T

H(u)–TH(u)_W,ϕ₍₎≤CuLϕ₍₎, (.)

whereis any bounded domain inRn,n≥,ϕ: [,∞)→[,∞) withϕ() =  is a Young function satisfying certain conditions described later, andCis a constant independent of the diﬀerential formu. In order to establish the above mainLϕ_{-embedding inequality,}

we also prove the Poincaré inequality and some inequalities withLϕ_{-norm for the related}

compositions of operators.

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We keep using the traditional notations throughout this paper. LetBandσBbe the balls with the same center anddiam(σB) =σdiam(B). Let|E|be then-dimensional Lebesgue measure of a setE⊆Rn_{. In this paper, we treat a ball same as a cube and use}_u

B=_|_B_|

Bu dx

to denote the average of a function u. Let ∧l₌_∧l₍_Rn_{) be the set of all}_l_{-forms in}_Rn_,

D(,∧l_{) be the space of all diﬀerential}_l_{-forms in}_{, and}_Lp₍_,_∧l_{) be the}_l_-forms_u₍_x_{) =}

IuI(x)dxIinsatisfying

|uI|

p_<_∞_{for all ordered}_l_-tuples_I_,_l_{= , , . . . ,}_n_{. We denote}

the exterior derivative bydand the Hodge star operator by.

The deﬁnition of the operatorKywith the casey=  and its generalized version can

be found in [, ]. To eachy∈there corresponds a linear operatorKy:C∞(,∧l)→

C∞(,∧l–_{) deﬁned by (}_K

yω)(x;ξ, . . . ,ξl–) =



tl–ω(tx+y–ty;x–y,ξ, . . . ,ξl–)dtand the

decompositionω=d(Kyω) +Ky(dω). A homotopy operatorT:C∞(,∧l)→C∞(,∧l–)

is deﬁned by averagingKyover all pointsy∈:Tω=

φ(y)Kyωdy, whereφ∈C∞ () is

normalized so thatφ(y)dy= . For each diﬀerential formu, we have the decomposition

u=d(Tu) +T(du) (.)

and

_∇(Tu)_p_,_B≤C|B|up,B and Tup,B≤C|B|diam(B)up,B. (.)

From [], p., we know that any open subsetinRn_{is the union of a sequence of cubes}

Qk, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose

diameters are approximately proportional to their distances fromF, whereFis the com-plement ofinRn_{. Speciﬁcally, (i)}₌∞

k=Qk, (ii)Qj∩Qk=φifj=k, (iii) there exist two

constantsc,c>  (we can takec= , andc= ), so thatcdiam(Qk)≤distance(Qk,F)≤

cdiam(Qk). Thus, the deﬁnition of the homotopy operatorT can be generalized to any

domaininRn_{: For any}_x_∈_,_x_∈_Q

k for somek. LetTQk be the homotopy operator

deﬁned onQk(each cube is bounded and convex). Thus, we can deﬁne the homotopy

op-eratorT on any domainbyT= _∞

k=TQkχQk(x). The nonlinear partial diﬀerential

equation for diﬀerential forms

dA(x,du) =B(x,du) (.)

is called a non-homogeneousA-harmonic equation, whereA:× ∧l_(Rn₎_{→ ∧}l_(Rn_{) and}

B:× ∧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– (.)

forx∈a.e. and allξ∈ ∧l_(Rn_{). Here}_p_{>  is a constant related to the equation (.), and}

a,b> . See [–] for recent results on theA-harmonic equations and related topics. As-sume that∧l_{is the}_l_{th exterior power of the cotangent bundle,}_C∞₍_∧l_{) is the space of}

smoothl-forms onandW(∧l_{) =}_{_u_∈_L

loc(∧l) :uhas generalized gradient}. The

har-monicl-ﬁelds are deﬁned byH(∧l_{) =}_{_u_∈_W₍_∧l_{) :}_du₌_d_u_{= ,}_u_∈_Lp_{for some  <}

p<∞}. The orthogonal complement ofHin L _{is deﬁned by}_H⊥₌_{_u_∈_L _:_u_,_h₌

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by assigningG(u) be the unique element ofH⊥∩C∞(∧l_{) satisfying Poisson’s equation}

G(u) =u–H(u), whereHis the harmonic projection operator that mapsC∞(∧l_{) onto}

Hso thatH(u) is the harmonic part ofu. See [] for more properties of these operators.

2 Local embedding theorem

The purpose of this section is to prove the localLϕ_{-embedding theorem and some related}

Lϕ_{-norm inequalities that will be used to prove the global embedding theorem in the next}

section. We ﬁrst recall the following subclass of Young functions that can be found in [–].

Deﬁnition . A Young functionϕ: [,∞)−→[,∞) is said to be in the classNG(p,q) if

ϕsatisﬁes the nonstandard growth condition

pϕ(t)≤tϕ(t)≤qϕ(t),  <p≤q<∞. (.)

The ﬁrst inequality in (.) is equivalent to thatϕ_t(pt)is increasing, and the second inequality

in (.) is equivalent to-condition,i.e., for eacht> ,ϕ(t)≤Kϕ(t), whereK> , and

ϕ(t)

tq is decreasing witht. Also, condition (.) implies thatϕ(t) satisﬁes ctp–c≤ϕ(t)≤c

tq+ . (.)

Particularly,ϕ(t) =tp _{satisﬁes (.) because of}_tϕ₍_t_{) =}_pϕ₍_t_{), and this makes inequalities}

with the norm · pbecome a special case of Theorem .; for more details see [] and

[].

An Orlicz function is a continuously increasing function ϕ : [,∞)→[,∞) with

ϕ() = . The Orlicz spaceLϕ₍_{) consists of all measurable functions}_f _on_{such that}

ϕ( |f|

λ)dx<∞for someλ=λ(f) > .L

ϕ₍_{) is equipped with the nonlinear Luxemburg}

functional

fLϕ₍₎=inf λ>  :

ϕ

|f| λ

dx≤

.

A convex Orlicz functionϕis often called a Young function. Ifϕis a Young function, then · Lϕ₍₎deﬁnes a norm inLϕ(), which is called the Luxemburg norm or Orlicz norm.

For any subsetE⊂Rn_{, we use}_W,ϕ₍_E_,_∧l_{) to denote the Orlicz-Sobolev space of}_l_-forms

which equalsLϕ₍_E_,_∧l₎_∩_Lϕ

(E,∧l) with norm

uW,ϕ₍_E₎=u_W,ϕ₍_E_,_∧l₎=diam(E)–uLϕ₍_E₎+∇u_Lϕ₍_E₎. (.)

If we chooseϕ(t) =tp_,_p_{>  in (.), we obtain the usual}_Lp_{-norm for}_W,p₍_E_,_∧l₎

uW,p₍_E₎=u_W,p₍_E_,_∧l₎=diam(E)–u_p_,_E+∇u_p_,_E. (.)

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Lemma .[] Let u∈D(B,∧l₎_{and du}_∈_Lp₍_B_,_∧l+_)._{Then u}_–_u

B∈L

np

n–p₍_B_,∧l_),_and

B

|u–uB|

np n–p_dx

n–p np

≤c(n,p)

B

|du|pdx 

p

(.)

for B is a ball or cube in,l= , , . . . ,n– and <p<n.

Lemma .[] Supposeϕis a continuous function in the class NG(p,q)with q(n–p) <np,

 <p≤q<∞.For any t> ,setting

A(t) =

t



ϕ(s/q₎

s n+q

q

ds, K(t) =(ϕ(t

/q₎₎n+qq

tn/q . (.)

Then A(t)is a concave function,and there exists a constant C,such that

K(t)≤A(t)≤CK(t), ∀t> . (.)

Lemma .[] Let u be a diﬀerential form satisfying the non-homogeneous A-harmonic

equation(.)in,σ> and <s,t<∞.Then,there exists a constant C,independent of u,such thatus,B≤C|B|(t–s)/stut,σBfor all balls or cubes B withσB⊂.

We are ready to state our main localLϕ_{-embedding theorem as follows, which will be}

used to prove the globalLϕ_{-embedding theorem in the next section.}

Theorem . Let ϕ be a Young function in the class NG(p,q)with q(n–p) <np,  <

p≤q<∞.be a bounded domain,u∈Lp₍_,_∧l₎_{be a solution of the non-homogeneous}

A-harmonic equation,T be the homotopy operator and H be the projection operator.If

ϕ(|u|)∈L_loc (),then there exists a constant C,independent of u,such that

TH(u)–TH(u)_B_W,ϕ₍_B_,_∧l₎≤C|B|uLϕ₍_σ_B₎ (.)

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

In order to prove the above localLϕ_{-embedding theorem, we need to prove some local}

Lϕ_{-norm inequalities. We begin with the following Poincaré-type inequality with}_Lϕ_-norm

ﬁrst.

Theorem . Letϕbe a Young function in the class NG(p,q)with q(n–p) <np,  <p≤ q<∞,be a bounded domain,u∈Lp₍_,_∧l_),_{T be the homotopy operator and H be the}

projection operator.Ifϕ(|u|)∈L

loc(),then there exists a constant C,independent of u,

such that

TH(u)–TH(u)_B_Lϕ₍_B₎≤C|B|uLϕ₍_B₎ (.)

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Proof First, we consider the case  <p<n. By assumption, we haveq< _nnp_–_p. Using the Poincaré-type inequality, Lemma . to diﬀerential formsT(H(u))

B

TH(u)–TH(u)_Bnp/(n–p)dx (n–p)/np

≤C

B

dTH(u)pdx /p

, (.)

we ﬁnd that

B

_T

H(u)–TH(u)_Bqdx /q

≤C

B

_d

TH(u)pdx /p

. (.)

It is well known that, for any diﬀerential formu,d(T(u)) =uB anduBp,B≤Cup,B.

Hence,

B

dTH(u)pdx /p

≤C

B

H(u)pdx /p

. (.)

Note that

_G(u)_p_,_B=dd+ddG(u)_p_,_B≤Cup,B.

We have

H(u)_p_,_B=u–G(u)_p_,_B ≤ up,B+G(u)_p_,_B

≤Cup,B. (.)

Combining (.), (.), and (.), we obtain

B

TH(u)–TH(u)_Bqdx /q

≤C

B

|u|pdx /p

(.)

for  <p<n. Next, for the case ofp≥n, since theLp-norm of|T(H(u)) – (T(H(u)))B|

increases withpand_nnp_–_p → ∞asp→n. Then, there exists  <p<nsuch thatq<_nnp_–_p_.

Hence, it follows that

B

≤

B

TH(u)–TH(u)_Bnp/(n–p)dx

(n–p)/np

≤C

B

dTH(u)p

dx /p

≤C

B

dTH(u)pdx /p

≤C

B

|u|pdx /p

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Hence, from (.) and (.), we obtain

B

≤C

B

|u|pdx /p

(.)

for anyp> . Using the Hölder inequality with  =_nq₊_q+ n

n+q, we obtain

B

ϕTH(u)–TH(u)_Bdx

=

B

ϕ(|T(H(u)) – (T(H(u)))B|)

|T(H(u)) – (T(H(u)))B|

nq n+q

TH(u)–TH(u)_B

nq n+q_dx

≤

B

ϕ(|T(H(u)) – (T(H(u)))B|)

n+q q

|T(H(u)) – (T(H(u)))B|n

dx q

n+q

×

B

TH(u)–TH(u)_Bqdx n

n+q

.

Applying Lemma . and noticingA(t) is a concave function, we obtain

B

≤

B

KTH(u)–TH(u)_Bqdx q

n+q

B

_T

H(u)–TH(u)_Bqdx n

n+q

≤

B

ATH(u)–TH(u)_Bqdx q

n+q

B

n+q

≤Anq+q

B

TH(u)–TH(u)_Bqdx

B

n+q

≤C(n,q)K

q n+q

B

TH(u)–TH(u)_Bqdx

×

B

n+q

=C(n,q)

ϕ((_B(|T(H(u)) – (T(H(u)))B|q)dx)/q)

(_B(|T(H(u)) – (T(H(u)))B|q)dx)

n n+q

×

B

n+q

=C(n,q)ϕ

B

. (.)

Note thatϕis increasing and satisﬁes-condition, substituting (.) into (.) gives

B

ϕTH(u)–TH(u)_Bdx≤Cϕ

B

|u|pdx /p

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Leth(t) =_tϕ(_ss)ds. From (.) we know thatϕ(t)/tq_{is decreasing with}_t_{, thus,}

h(t) =

t



ϕ(s)

s ds=

t



ϕ(s)

sq s

q–_ds_≥_ϕ₍_t_)/_tq

qs

q t



=

qϕ(t).

Similarly, using the fact thatϕ(t)/tp_{is increasing with}_t_{, we have}_h₍_t₎_≤ 

pϕ(t). Therefore,



qϕ(t)≤h(t)≤



pϕ(t). (.)

Letg(t) =h(t/p_{), then (}_h₍_t/p₎₎₌

p

ϕ(t/p)

t is increasing. Hence,gis a convex function. From

the deﬁnitions ofgandhand using Jensen’s inequality tog, we have

h

B|

u|pdx /p

=g

B|

u|pdx

≤

B

g|u|pdx=

B

h|u|dx. (.)

Combining (.), (.), and (.), we have

B

≤Cϕ

B

|u|pdx /p

≤Ch

B

|u|pdx /p

≤C

B

h|u|dx

≤C

B

ϕ|u|dx,

which indicates (.) holds. We have completed the proof of Theorem ..

Theorem . Let ϕ be a Young function in the class NG(p,q)with q(n–p) <np,  <

p≤q<∞,be a bounded domain,u∈Lp₍_,_∧l₎_{be a solution of the non-homogeneous}

ϕ(|u|)∈L_loc (),then there exists a constant C,independent of u,such that

TdTH(u)_Lϕ₍_B₎≤C|B|uLϕ₍_σ_B₎

Proof For any diﬀerential formu, we havedT(u)q,B=uBq,B≤Cuq,B. From (.)

and (.), it follows that

TdTH(u)_q_,_B≤C|B|diam(B)dTH(u)_q_,_B

=C|B|diam(B)H(u)_q_,_B

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By Lemma . andp,q> , we have

uq,B≤C|B|(p–q)/pqup,σB, (.)

whereσ> . Combining (.) and (.) yields

TdTH(u)_q_,_B≤C|B|(p–q)/pqup,σB. (.)

From the Hölder inequality with  =_n₊q_q+_nn₊_q, we ﬁnd that

B

ϕTdTH(u)dx

=

B

ϕTdTH(u)TdTH(u)–

nq

n+q_TdTH₍_u₎ nq n+q_dx

≤

B

ϕ(|TdTH(u)|)

n+q q

|TdTH(u)|n dx

q n+q

B

TdTH(u)qdx n

n+q

.

Using Lemma . and noticingA(t) is a concave function, we obtain

B

ϕTdTH(u)dx

≤

B

KTdTH(u)qdx q

n+q

B

TdTH(u)qdx n

n+q

≤

B

ATdTH(u)qdx q

n+q

B

_TdTH₍_u₎q

dx n

n+q

≤Anq+q

B

TdTH(u)qdx

B

TdTH(u)qdx n

n+q

≤CK

q n+q

B

TdTH(u)qdx

B

TdTH(u)qdx n

n+q

=C

ϕ((_B(TdTH(u)q)dx)/q₎

(_B(|TdTH(u)|q₎_dx₎n+nq

B

TdTH(u)qdx n

n+q

=Cϕ

B

TdTH(u)qdx /q

. (.)

Sinceϕis increasing and satisﬁes-condition, substituting (.) into (.), we have

B

ϕTdTH(u)dx≤Cϕ

σB

|u|pdx /p

. (.)

Starting from (.) and using the same discussion as we did in the proof of Theorem ., we obtain

B

ϕTdTH(u)dx≤C

σB

ϕ|u|dx.

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From (.) and (.), we have

_∇_TdTH(u)_q_,_B≤C|B|dTH(u)_q_,_B

=C|B|H(u)

Bq,B

≤C|B|H(u)_q_,_B

≤C|B|uq,B. (.)

Using (.) and the similar techniques to the ones developed in the proof of Theorem ., we obtain the followingLϕ_{-norm estimate.}

Theorem . Letϕbe a Young function in the class NG(p,q)with q(n–p) <np,  <p≤ q<∞,be a bounded domain,and u∈Lp₍_,_∧l₎_{be a solution of the non-homogeneous}

ϕ(|u|)∈L

loc(),then there exists a constant C,independent of u,such that

_∇TdTH(u)_Lϕ₍_B₎≤C|B|uLϕ₍_σ_B₎ (.)

Proof of Theorem. By deﬁnition (.), Theorem . and Theorem ., we have

TH(u)–TH(u)_B_W,ϕ₍_B_,_∧l₎

=TdTH(u)_W,ϕ₍_B_,_∧l₎

=diam(B)–TdTH(u)_Lϕ₍_B₎+∇TdTH(u)_Lϕ₍_B₎

≤diam(B)–C|B|diam(B)uLϕ₍_σ__B₎+C_|B|u_Lϕ₍_σ__B₎

≤C|B|uLϕ₍_σ_B₎, (.)

whereσ=max{σ,σ}. We have completed the proof of Theorem ..

3 Global embedding theorem

In this section, we prove our main result, the globalLϕ_{-embedding theorem for the}

so-lutions of the non-homogeneousA-harmonic equation. We will use the following well-known covering lemma.

Lemma . Each domainhas a modiﬁed Whitney cover of cubesV={Qi}such that

i

Qi=,

Qi∈V χ

Qi≤Nχ

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

member ofV)in Qi∩Qjsuch that Qi∪Qj⊂NR.Moreover,ifisδ-John,then there is a

distinguished cube Q∈Vwhich can be connected with every cube Q∈Vby a chain of cubes

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We are ready to prove the following globalLϕ_{-embedding theorem with the}_Lϕ_-norm

now.

Theorem . Letϕbe a Young function in the class NG(p,q)with q(n–p) <np,  <p≤

q<∞,u∈Lp₍_,_∧l₎_{be a solution of the non-homogeneous A-harmonic equation}_,_{T be the}

homotopy operator and H be the projection operator.Ifϕ(|u|)∈L₍_),_{then there exists a}

constant C,independent of u,such that

TH(u)–TH(u)_W,ϕ₍_,_∧l₎≤CuLϕ₍₎ (.)

for a bounded domain⊂Rn_.

Proof From the Lemma . and Theorem ., we have

_∇TdTH(u)_Lϕ₍₎≤

B∈V

_∇TdTH(u)_Lϕ₍_B₎

≤

B∈V

C|B|uLϕ₍_σ_B₎

≤CNuLϕ₍₎

≤CuLϕ₍₎. (.)

Similarly, from Theorem . and Lemma ., it follows that

_Td

TH(u)_Lϕ₍₎≤

B∈V _Td

TH(u)_Lϕ₍_B₎

≤

B∈V

Cdiam(B)uLϕ₍_σ_B₎

≤Cdiam()NuLϕ₍₎

≤Cdiam()uLϕ₍₎. (.)

Using (.), (.), and (.), we ﬁnd that

TH(u)–TH(u)_W,ϕ₍₎

=TdTH(u)_W,ϕ₍₎

=diam()–TdTH(u)_Lϕ₍₎+∇Td

TH(u)_Lϕ₍₎

≤diam()–Cdiam()uLϕ₍₎+C_u_Lϕ₍₎

≤CuLϕ₍₎. (.)

We have completed the proof of Theorem ..

Choosingϕ(t) =tp_logα

+tin Theorems ., we have the following embedding inequality

with theLp_(logα

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Corollary . Letϕ(t) =tp_logα

+t, p≥,α∈R, u∈Lp(,∧l)be a solution of the

non-homogeneous A-harmonic equation,T be the homotopy operator,and H be the projection

operator.Ifϕ(|u|)∈L₍_),_{then there exists a constant C}_,_{independent of u}_,_{such that}

TH(u)–TH(u)_W,tplogα₊t₍₎≤Cu_Ltplogα+t₍₎ (.)

for any bounded domain.

Letϕ(t) =tp _{in Theorem .. Then, we obtain the following version of the embedding}

inequality withLp_-norms.

Corollary . Letϕ(t) =tp, p≥,u∈Lp(,∧l)be a solution of the non-homogeneous A-harmonic equation,T be the homotopy operator,and H be the projection operator.If ϕ(|u|)∈L₍_),_{then there exists a constant C}_,_{independent of u}_,_{such that}

TH(u)–TH(u)_W,p₍₎≤Cup,

holds for any bounded domain.

Similarly, Theorem . can be extended into the following global Poincaré-type inequal-ity withLϕ_-norm.

Theorem . Letϕbe a Young function in the class NG(p,q)with q(n–p) <np,  <p≤ q<∞,u∈Lp₍_,_∧l₎_{be a solution of the non-homogeneous A-harmonic equation}_,_{T be the}

homotopy operator and H be the projection operator.Ifϕ(|u|)∈L₍_),_{then there exists a}

constant C,independent of u,such that

TH(u)–TH(u)_Lϕ₍₎≤Cdiam()uLϕ₍₎ (.)

for any bounded domain.

Proof From (.) and Theorem ., we have

TH(u)–TH(u)_Lϕ₍₎

=diam()diam()–TH(u)–TH(u)_Lϕ₍₎

≤diam()diam()–TH(u)–TH(u)_Lϕ₍₎

+∇TH(u)–TH(u)_Lϕ₍₎

=Cdiam()T

H(u)–TH(u)_W,ϕ₍_,_∧l₎

≤Cdiam()uLϕ₍₎.

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4 Applications

As applications of our main results established in the previous sections, we consider the following examples.

Example . Assume thatr>  andk>  are any constants and={(x,x,x) :x+x+

x

≤r} ⊂R. Consider the -form

u(x,x,x) =

x

k+x

+x+x

dx+

x

k+x

+x+x

dx+

x

k+x

+x+x

dx (.)

deﬁned in. It is easy to check thatdu= . Hence,uis a solution of the non-homogeneous

A-harmonic equation (.) for any operatorsAandBsatisfying (.). Also, it can be cal-culated that

|u|=

x

k+x_+x_+x_ 

+

x

k+x_+x_+x_ 

+

x

k+x_+x_+x_ /

=

x

+x+x

(k+x

+x+x)

/

< . (.)

Using (.) and (.), we ﬁnd that

TH(u)–TH(u)_W,ϕ₍₎≤CuLϕ₍₎≤C__Lϕ₍₎,

that is,

TH(u)–TH(u)_W,ϕ₍₎≤C

ϕ



λ

dx≤Cr.

We should notice that the above example can be extended to the case ofRn_{. Speciﬁcally,}

we can check that the -form deﬁned inRn

u(x, . . . ,xn) = n

i=

xi

k+x_+· · ·+x

n

dxi, k>  (.)

is a solution of the non-homogeneousA-harmonic equation (.) for any operatorsAand

Bsatisfying (.). Hence, Theorem . is applicable tou(x, . . . ,xn) as we did in Example ..

Finally, we consider the following example inR_.

Example . Let={(x,y,z) :x+y+z≤} ⊂Randu(x,y,z) be deﬁned inRby

u(x,y,z) =ex+y+z(x dx+y dy+z dz).

It is easy to check thatdu= . Hence,uis a solution of the non-homogeneousA-harmonic equation (.) for any operators A and Bsatisfying (.) in R_{. For any bounded}

do-main in R_{, it would be very complicated if we calculate the integral} _T₍_H₍_u_{)) –}

(T(H(u)))W,ϕ₍₎directly. However, using the embedding inequality (.), we can easily

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that

u(x,y,z)=ex+y+zx+ex+y+zy+ex+y+zz/

=e(x+y+z)x+y+z/

≤e·/=e. (.)

From (.) and (.), we have

TH(u)–TH(u)_W,ϕ₍₎≤CuLϕ₍₎≤C_e_Lϕ₍₎,

which is equal to

TH(u)–TH(u)_W,ϕ₍₎≤C

ϕ

e λ

dx≤C.

Remark (i) From inequalities (.), (.), and (.), we ﬁnd that the compositionsTH,

TdTH and∇TdTH are bounded operators onLp₍_,_∧l_{). (ii) Note that our global}

embed-ding theorem holds on any bounded domains. Hence, the theorem is true ifis one of the bounded John domains, Lp_{-averaging domains or}_Lϕ₍_μ_{)-averaging domains. See [,}

, ] for more properties of these kinds of domains.

Competing interests

The authors declare that there is no conﬂict of interests regarding the publication of this article.

Authors’ contributions

All authors worked together on the research and writing of this manuscript. JN carried out the proofs of all research results in this manuscript, and wrote its draft. SD and YX proposed the study, participated in its design and revised its ﬁnal version. All authors read and approved the ﬁnal manuscript.

Author details

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

Seattle University, Seattle, WA 98122, USA.

Received: 14 August 2015 Accepted: 1 November 2015

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