Inequalities for the composition of Green’s operator and the potential operator

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R E S E A R C H

Open Access

Inequalities for the composition of Green’s

operator and the potential operator

Zhimin Dai

1*

_{, Yuming Xing}

1

_{, Shusen Ding}

2

_{and Yong Wang}

1

*_{Correspondence: zmdai@yahoo.cn} 1_{Department of Mathematics,}

Harbin Institute of Technology, Harbin, China

Full list of author information is available at the end of the article

Abstract

We ﬁrst establish theLp-norm inequalities for the composition of Green’s operator and the potential operator. Then we develop theLϕ-norm inequalities for the composition in theLϕ-averaging domains. Finally, we display some examples for applications.

MSC: Primary 35J60; secondary 31B05; 58A10; 46E35

Keywords: diﬀerential forms;Lp_{-norm; the Luxemburg norm; Green’s operator; the} potential operator

1 Introduction

The purpose of this paper is to derive some inequalities for the composition of Green’s op-eratorGand the potential operatorPapplied to differential forms. The differential forms are extensions of functions and can be used to describe various systems of PDEs, physics, theory of elasticity, quasiconformal analysis,etc.In the meanwhile, Green’s operator and the potential operator are of considerable importance in the study of potential theory and nonlinear elasticity; see [–] for more properties of these two operators. In many situa-tions, the process to study solutions of PDEs involves estimating the various norms of the operators. Bi defined a potential operatorPapplied to differential forms in []. However, the study on the composition of the potential operator and other operators has just begun. Hence, we are motivated to establish some inequalities for the composite operatorG◦P

applied to diﬀerential forms.

Now we introduce some notations. Unless otherwise indicated, we always use to denote an open subset of Rn ₍_n_≥_{), and let} _O _{be a ball in} _Rn_{. Let} _ρ_O _{denote the} ball with the same center asO and diam(ρO) =ρdiam(O), ρ> . A weight w(x) is a nonnegative locally integrable function inRn_._|_D_|_{is used to denote the Lebesgue} mea-sure of a setD⊂Rn_{. Let}_∧₌_∧₍_Rn_), _{= , , . . . ,}_n_{, be the linear space of all}_-forms

(x) =_JJ(x)dxJ =

Jjj···j(x)dxj∧dxj ∧ · · · ∧dxj inR

n_{, where}_J_{= (}_j

,j, . . . ,j), ≤j<j<· · ·<j≤n, are the ordered-tuples. Moreover, if each of the coefficientsJ(x) of(x) is differential on, then we call(x) a differential-form onand useD(,∧_{) to} denote the space of all differential-forms on.C∞(,∧_{) denotes the space of smooth}

-forms on. The exterior derivatived:D(,∧₎_→_D₍_,_∧+_),_{= , , . . . ,}_n_{–, is given}

by

d(x) = n

i=

J

∂jj···j(x)

∂xi

dxi∧dxj∧dxj∧ · · · ∧dxj (.)

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for all∈D(,_{) and the Hodge codiﬀerential operator}_d_{is deﬁned as}_d_{= (–)}n+

d:D(,+₎_→_D₍_,_{), where}_{is the Hodge star operator.}_Lp₍_,_∧_{) is a Banach} space with the norm

p,=

(x)pdx

/p =

J

_J(x)

p/

dx

/p

<∞. (.)

For a weightw(x), we writep,,w= (

||

p_w₍_x₎_dx₎/p_{. From [], if}_{is a diﬀerential form}

in a bounded convex domain, then there is a decomposition

=d(T) +T(d), (.)

where T is called a homotopy operator. Furthermore, we can deﬁne the-form ∈

D(,∧_{) by}

=

⎧ ⎨ ⎩

||–(y)dy, = ;

d(T), = , . . . ,n. (.)

for all∈Lp₍_,_∧_{), }_≤_p_<_∞_.

With respect to the nonhomogeneousA-harmonic equation for diﬀerential forms, we indicate the general form as follows:

d_A₍_x_,_d_{) =}_B₍_x_,_d_), _(.)

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

|A(x,η)| ≤a|η|p–,A(x,η)·η≥ |η|pand|B(x,η)| ≤b|η|p–for almost everyx∈and all

η∈ ∧₍_Rn_{). Here}_a_,_b_{>  are some constants and  <}_p_<_∞_{is a ﬁxed exponent associated} with (.). A solution to (.) is an element of the Sobolev spaceW_loc,p(,∧–_{) such that}

A(x,d)·dψ+B(x,d)·ψ=  (.)

for all ψ ∈W_loc,p(,∧–_{) with compact support, where} _W,p

loc(,∧–) is the space of -forms whose coeﬃcients are in the Sobolev spaceW_loc,p().

Recently, Bi extended the definition of the potential operator to the set of all differential forms in []. For any differential -form(x) =_JJ(x)dxJ, the potential operator Pis defined by

P(x) =P

J

J(x)dxJ

=

J

PJ(x)

dxJ=

J

K(x,y)J(y)dy dxJ, (.)

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2 TheLp_{-norm inequalities}

In this section, we establish theLp_{-norm inequality for the composite operator}_G_◦_P_and obtain theAr,λ()-weight version of the inequality. We need the following deﬁnitions and lemmas.

In [], Ding gives the followingLp_{-norm inequality for Green’s operator.}

Lemma . Let∈Lp(,∧₎_{be a smooth form in a bounded domain}_,_{= , . . . ,}_n_{,  <}

p<∞and G be Green’s operator.Then there exists a constant C,independent of,such that

G() –G()_O_p_,_O≤C|O|diam(O)p,O (.)

for all balls O⊂.

Deﬁnition .([]) A pair of weights (w(x),w(x)) satisﬁes theAr,λ()-condition in a set

⊂Rn, write (w(x),w(x))∈Ar,λ() for someλ≥ and  <r<∞with _r+_r = , if

sup

O⊂



|O|

O

wλ_dx



λr _

|O|

O



w

λr r

dx



λr

<∞. (.)

The following deﬁnition is introduced in [].

Deﬁnition . A kernelK onRn×Rn(n≥) satisﬁes the standard estimates if there existα,  <α≤, and a constantCsuch that, for all distinct pointsxandyinRn_{and all}_z with|x–z|<_|x–y|:

() K(x,y)≤C|x–y|–n; () K(x,y) –K(z,y)≤Cx–z

x–y

α|x–y|–n;

() K(z,y) –K(x,y)≤Cx–z x–y

α|x–y|–n.

(.)

The following two-weight norm inequality for the potential operatorPapplied to dif-ferential forms appears in [].

Lemma .([]) Let∈Lp(,∧_), _{= , , . . . ,}_n_{,  <}_p_<_∞_,_{be a diﬀerential form in a}

domainand P be the potential operator deﬁned in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.).Assume that(w,w)∈Ar,λ()for some

λ≥and <r<∞.Then there exists a constant C,independent of,such that

P()_p_,_,_w

≤Cp,,w. (.)

Remark If letw=w=  in (.), then (.) reduces to the following inequality:

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Theorem . Let ∈ Lp₍_,_∧_), _{= , , . . . ,}_n_{,  <} _p_<_∞_, _{be a diﬀerential form in a}

bounded convex domain⊂Rn,P be the potential operator deﬁned in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.

Then there exists a constant C,independent of,such that

GP()–GP()_O_p_,_O≤C|O|diam(O)p,O (.)

for all balls O with O⊂.

Proof By Lemma . and (.), we have

GP()–GP()_O_p_,_O≤C|O|diam(O)P()_p_,_O

≤C|O|diam(O)p,O. (.)

We complete the proof of Theorem ..

Lemma . Let <α,β<∞and/s= /α+ /β.If f and g are two measurable functions onRn_,_then

fgs,≤ fα,· gβ, (.)

for any⊂Rn.

Lemma .([]) Letbe a solution of the nonhomogeneous A-harmonic equation(.)

in a domainand <s,t<∞.Then there exists a constant C,independent of,such that

s,O≤C|O|(t–s)/stt,ρO (.)

for all balls O withρO⊂,whereρ> is a constant.

Based on Theorem ., we obtain the following Ar,λ()-weight version of inequal-ity (.).

Theorem . Let∈Lp₍_,_∧_), _{= , , . . . ,}_n_{,  <}_p_<_∞_, _{be a solution of the}

nonho-mogeneous A-harmonic equation (.)in a bounded convex domain,P be the poten-tial operator deﬁned in (.) with the kernel K(x,y) satisfying the condition () of the standard estimates(.)and G be Green’s operator.Assume that(w(x),w(x))∈Ar,λ()

for some λ≥ and <r<∞.Then there exists a constant C, independent of , such that

GP()–GP()_O_p_,_O_,_wα

 ≤C|O|

diam(O)p,ρO,wα_ (.)

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Proof Lets=λp/(λ–α) andt=λp/(λ+α(r– )), thus  <t<p<s. Using Theorem ., Lemma . with /p= /s+ (s–p)/(sp) and Lemma ., we obtain

GP()–GP()_O_p_,_O_,_wα



=

O

GP()–GP()_Opwα_dx

/p

=

O

_G

P()–GP()_Owα_/ppdx

/p

≤

O

_G

P()–GP()_Osdx

/s

O

wsα/(s–p)dx

(s–p)/(sp)

=GP()–GP()_O_s_,_O

O

wλ

dx

α/(λp)

≤C|O|diam(O)s,O

O

wλ

dx

α/(λp)

≤C|O|diam(O)|O|(t–s)/(st)t,ρO

O

wλ

dx

α/(λp)

. (.)

Applying Lemma . with /t= /p+ (p–t)/(pt) yields

t,ρO=

ρO

||t_dx

/t =

ρO

||wα_/pw–_α/ptdx

/t

≤

ρO

||p_wα

dx /p ρO  w

αt/(p–t)

dx

(p–t)/(pt)

=p,ρO,wα

ρO



w

λ/(r–)

dx

α(r–)/(λp)

. (.)

Note that (w(x),w(x))∈Ar,λ(), therefore

O

wλ

dx

α/(λp)

ρO



w

λ/(r–)

dx

α(r–)/(λp)

≤ |ρO|α/(λp)+α(r–)/(λp)



|ρO|

ρO

wλ_dx



λr _

|ρO|

ρO



w

λ/(r–)

dx

r–

λrαr/p

≤C|ρO|α/(λp)+α(r–)/(λp)

≤C|O|α/(λp)+α(r–)/(λp). (.)

It is easy to check that (t–s)/(st) +α/(λp) +α(r– )/(λp) = , thus

GP()–GP()_O_p_,_O_,_wα

 ≤C|O|

diam(O)|O|(t–s)/(st)+α/(λp)+α(r–)/(λp)

p,ρO,wα_

=C|O|diam(O)p,ρO,wα

. (.)

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Deﬁnition . The weightw(x) is said to satisfy theAr() condition,r> . Writew∈

Ar() ifw(x) >  a.e. and

sup

O



|O|

O

w dx



|O|

O



w



r– dx

(r–)

<∞ (.)

for any ballO⊂.

Ifλ= ,w=w, theAr,λ()-weights reduce to the usual class ofAr()-weights; see [] for more details about weights. So, we have the following corollary.

Corollary . Let∈Lp(,∧_),_{= , , . . . ,}_n_{,  <}_p_<_∞_, _{be a solution of the}

nonho-mogeneous A-harmonic equation(.)in a bounded convex domain,P be the potential operator deﬁned in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.Assume that w(x)∈Ar()for <r<∞.Then

there exists a constant C,independent of,such that

GP()–GP()_O_p_,_O_,_wα≤C|O|diam(O)p,ρO,wα (.)

for all balls O withρO⊂,whereρ> andαare two constants with≤α< .

3 TheLϕ-norm inequalities

In this section, we ﬁrst recall some deﬁnitions of elementary conceptions, including the Luxemburg norm and the classG(p,q,C) of Young functions. Then we prove theLϕ_-norm inequalities for the composite operatorG◦P.

A continuously increasing functionϕ: [,∞)→[,∞) withϕ() =  andϕ(∞) =∞is called an Orlicz function, and a convex Orlicz function is often called a Young function. The Orlicz spaceLϕ₍_{) consists of all measurable functions}_f_on_{such that}

ϕ(

|f| χ)dx<

∞for someχ=χ(f) > , then the nonlinear Luxemburg functional off is denoted by

fϕ()=inf

χ>  :

ϕ

_|

f|

χ

dx≤

. (.)

Ifϕis a Young function, then · ϕ()deﬁnes a norm inLϕ(), which is called the

Luxem-burg norm.

The following class G(p,q,C) is introduced in [], which is a special class of Young functions.

Deﬁnition . We call a Young functionϕbelongs to the classG(p,q,C), ≤p<q<∞,

C≥, if

(i) 

C≤

ϕ(tp₎

f(t) ≤C; (ii) 

C≤

ϕ(tq₎

g(t) ≤C (.)

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From [], we assert thatϕ,f,g in the above deﬁnition are doubling, namelyϕ(t)≤

Cϕ(t) for allt> , and the completely similar property remains valid ifϕis replaced

cor-respondingly withf,g. Besides, we have

(i) Ctp≤f–

ϕ(t)≤Ctp; (ii) Ctq≤g–

ϕ(t)≤Ctq, (.)

whereC,CandCare some positive constants.

Bi [] constructs a special kernel function of a potential operator. Suppose the function

φ(x) is deﬁned as follows:

φ(x) =

⎧ ⎨ ⎩



cexp{



|x|_–}, |x|< ,

, |x| ≥,

(.)

wherec=_B_(,)e 

|x|–_dx_{. For any}_ε_{> , we write}_φ_ε₍_x_{) =} 

εnφ(x_ε). It is easy to see thatφ(x)∈

C_∞(Rn_{) and}

Rn

φε(x)dx= . (.)

LetPbe the potential operator in (.) withK(x,y) =φε(x–y). Assume that∈D(,∧),

= , , . . . ,n– , is a differential form defined in a bounded convex domainandJ is the coefficient ofwithsupp_J⊂for all ordered-tuplesJ. From (.) in [], we have the inequality as follows:

dP()_p_,≤Cp,, (.)

whereCis a constant, independent of, andpis a positive number with  <p<∞. Now we introduce two lemmas which will be needed later.

Lemma .([]) Let∈D(O,∧_),_{= , , . . . ,}_n_{– ,  <}_p_<_n_,_{and d}_∈_Lp₍_O_,_∧+_)._Then –Ois in L

np

n–p₍_O_,∧₎_and

O

|–O| np n–p

n–p np

≤Cp(n)

O

|d|p

/p

(.)

for O a cube or a ball inRn_.

Lemma .([]) Let∈C∞(,∧_),_{= , , . . . ,}_n_{–,  <}_p_<_∞_,_{and G be Green’s operator}_.

Then there exists a positive constant C,independent of,such that

ddG()_p_,+ddG()_p_,+dG()_p_,+dG()_p_,+G()_p_,

≤Cp, (.)

for any⊂Rn.

Theorem . Let∈D(,∧_),_{= , , . . . ,}_n_{– ,}_{be a diﬀerential form in a bounded}

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Assume that ϕ is a Young function in the class G(p,q,C), ≤p<q<∞, C≥ and q(n–p) <np,P is the potential operator in(.)with K(x,y) =φε(x–y)for anyε> and

G is Green’s operator.Then there exists a constant C,independent of,such that

O

ϕGP()–GP()_Odx≤C

O

ϕ||dx (.)

Proof Using Jensen’s inequality forg–_{that is deﬁned in Deﬁnition ., (.), (.) and}

noticing thatϕandgare doubling, for any ballO⊂, we obtain

O

ϕGP()–GP()_Odx

=g

g–

O

≤g

O

g–ϕGP()–GP()_Odx

≤g

C

O

_G

P()–GP()_Oqdx

≤Cϕ

C

O

GP()–GP()_Oqdx



q

≤Cϕ

O

GP()–GP()_Oqdx



q

. (.)

By Lemma ., it follows that

G()_p_,_O≤Cp,O. (.)

If ≤p<n, by assumption, we haveq< _nnp_–_p. Noticing that theLp_{-norm of}_|_G₍_P₍_{)) –} (G(P()))O|increases withp, using Lemma . forG(P()), (.) and (.), we have

O

GP()–GP()_Oqdx



q

≤C

O

GP()–GP()_O np n–p_dx

n–p np

≤C

O

dGP()pdx



p

=C

O

GdP()pdx



p

≤C

O

dP()pdx



p

≤C

O

p

dx



p

(.)

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Whenp≥n, we could select a strictly increasing sequence{pi}with ≤pi<n≤pand

pi→nasi→ ∞. Noticing that_nnp_–_pi

i→ ∞withi→ ∞, by keeping order of the limit, there existsiwith ≤i<∞such thatp<q<

npi

n–pi_, then

O

GP()–GP()_Oqdx



q

≤C

O

GP()–GP()_O npi

n–_pi  _dx

n–_pi 

npi

≤C

O

||pi_dx



pi

≤C

O

||p_dx



p

. (.)

Thus, (.) holds for anyp,qwith ≤p<q<∞,q(n–p) <np. Sinceϕis increasing, from (.) and (.), we obtain

O

ϕGP()–GP()_Odx≤Cϕ

C

O

||p_dx



p

. (.)

Applying (.), (i) in Deﬁnition ., Jensen’s inequality, and noticing thatϕandf are dou-bling, we have

O

ϕGP()–GP()_Odx≤Cϕ

C

O

||p_dx



p

≤Cf

C

O

||p_dx

≤C

O

f||pdx

≤C

O

ϕ||dx. (.)

Therefore, the proof of Theorem . has been completed.

Since each ofϕ,f andgin Deﬁnition . is doubling, from the proof of Theorem ., we have

O

ϕ

_|

G(P()) – (G(P()))O|

χ

dx≤C

O

ϕ

_|_|

χ

dx (.)

for all ballsOwithO⊂and any constantχ> . From the deﬁnition of the Luxemburg norm and (.), the following inequality with the Luxemburg norm

GP()–GP()_O_ϕ₍_O₎≤Cϕ(O) (.)

holds under the conditions described in Theorem ..

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belongs toG(p,p,C), ≤p<p<p,t>  andα∈R. Herelog+tis a piecewise function

such thatlog₊t=  fort≤e; otherwise,log₊t=logt. Moreover, ifα= , one veriﬁes easily that ϕ(t) =tp _{is as well in the class}_G₍_p

,p,C), ≤p<p <∞. Therefore, ﬁxing the

functionϕ(t) =tp_logα

+t,α∈Rin Theorem ., we obtain the following result.

Corollary . Let∈D(,∧_), _{= , , . . . ,}_n_{– ,}_{be a diﬀerential form in a bounded}

convex domain,andJ be the coeﬃcient ofwithsuppJ⊂for all ordered-tuples J.

Assume thatϕ=tp_logα

+t, ≤p<q<∞,C≥and q(n–p) <np,P is the potential operator

in(.)with K(x,y) =φε(x–y)for anyε> and G is Green’s operator.Then there exists a

constant C,independent of,such that

O

_G

P()–GP()_Oplogα₊GP()–GP()_Odx

≤C

O|| p_logα

+||dx (.)

4 Global inequalities

In this section, we ﬁrst recall the deﬁnition of theLϕ_{-averaging domains. Then we extend} the localLϕ_{-norm inequality for the composite operator}_G_◦_P_{to the global case in this} kind of domains.

Deﬁnition .([]) Letϕbe a Young function on [, +∞) withϕ() = . A proper sub-domain⊂Rn_{is called an}_Lϕ_{-averaging domain if}_|_|_<_∞_{, and there exists a constant}

Csuch that

ϕτ|–O|

dx≤Csup

O⊂

O

ϕσ|–O|

dx (.)

for some ballO⊂and all functionssuch thatϕ(||)∈Lloc(), whereτ,σ are

con-stants with  <τ,σ≤ ∞, and the supremum is over all ballsOwithO⊂.

Theorem . Let∈D(,∧₎_and_{be a bounded convex L}ϕ_{-averaging domain with}

supp⊂.Assume thatϕis a Young function in the class G(p,q,C), ≤p<q<∞,C≥

and q(n–p) <np,P is the potential operator in(.)with K(x,y) =φε(x–y)for anyε> 

and G is Green’s operator.Then there exists a constant C,independent of,such that

ϕGP()–GP()_O

dx≤C

ϕ||dx, (.)

where O⊂is some ﬁxed ball.

Proof Note thatis anLϕ_{-averaging domain and}_ϕ_{is doubling. From Deﬁnition . and} (.), we have

ϕGP()–GP()_O

dx≤Csup

O⊂

O

≤Csup

O⊂

C

O

ϕ||dx

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≤Csup

O⊂

C

ϕ||dx

≤C

ϕ||dx. (.)

We have completed the proof of Theorem ..

Similarly, by (.), we conclude that

GP()–GP()_O

ϕ()≤Cϕ(). (.)

From the deﬁnition of theLϕ_{-averaging domains, we see that an}_Ls_{-averaging domain []} is a specialLϕ_{-averaging domain when}_ϕ₍_t_{) =}_ts_{in Deﬁnition .. Hence, we have the} fol-lowing result in theLs_{-averaging domains.}

Corollary . Let∈D(,∧₎_and_{be a bounded convex L}s_{-averaging domain with}

supp⊂.Assume thatϕis a Young function in the class G(p,q,C), ≤p<q<∞,C≥

and q(n–p) <np,P is the potential operator in(.)with K(x,y) =φε(x–y)for anyε> 

and G is Green’s operator.Then there exists a constant C,independent of,such that

ϕGP()–GP()_O

dx≤C

ϕ||dx, (.)

where O⊂is some ﬁxed ball.

5 Applications

In this section, we give some examples of applications. By Theorem ., we can obtain other norm inequalities for the composite operatorG◦P, such as Lipschitz and BMO norms. Now, we take the Lipschitz norm for example.

Deﬁnition .([]) Let∈L

loc(,∧),= , , . . . ,n. We write∈locLipk(,∧), ≤

k≤, if

locLipk,= sup ρO⊂|

O|–(n+k)/n–O,O<∞ (.)

for someρ≥. Further, we write Lip_k(,∧_{) for those forms whose coeﬃcients are in the} usual Lipschitz space with exponentkand writeLipk,for this norm.

Theorem . Let ∈ Lp₍_,_∧_), _{= , , . . . ,}_n_{,  <} _p_<_∞_, _{be a diﬀerential form in a}

bounded domain, P be the potential operator deﬁned in(.)with the kernel K(x,y)

satisfying the condition()of the standard estimates(.)and G be Green’s operator.Then there exists a constant C,independent of,such that

GP()_locLip

k,≤Cp,, (.)

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Proof From Theorem ., we have

GP()–GP()_O_p_,_O≤C|O|diam(O)p,O (.)

for all ballsOwithO⊂. Using the Hölder inequality with  = /p+ (p– )/p, we get that

_G

P()–GP()_O_,_O=

O

_G

P()–GP()_Odx

≤

O

GP()–GP()_Opdx

/p

O

p/(p–)dx

(p–)/p

=|O|–/pGP()–GP()_O_p_,_O =|O|–/pC|O|diam(O)p,O

≤C|O|–/p+/np,O, (.)

where we have useddiam(O) =C|O|/n_{. Note that the deﬁnition of Lipschitz norm, (.)} and  – /p+ /n–  –k/n=  – /p+ /n–k/n>  yield

GP()_locLip

k, =_ρsup_O_⊂|O|

–(n+k)/n_G_P₍₎_–_G_P₍₎ O,O

≤ sup

ρO⊂

|O|––k/nC|O|–/p+/np,O

= sup

ρO⊂

C|O|–/p+/n–k/np,O

≤ sup

ρO⊂

C||–/p+/n–k/np,O

≤C sup

ρO⊂ p,O

≤Cp,. (.)

Thus, we have ﬁnished the proof of Theorem ..

Next, we would like to use Theorem . to make some estimate.

Example . Forn≥, letbe a -form deﬁned inRn_by

= x

x

+x+· · ·+xn

dx+

x

+x+· · ·+xn

dx

+· · ·+ xn

x

+x+· · ·+xn

dxn (.)

andϕ(t) =tplog₊t(t> ), wherex= (x, . . . ,xn)∈O⊂Rn\(, . . . , ). Green’s operatorG and the potential operatorPare deﬁned as in Theorem ..

It is easy to get that ||=  and ϕ belongs to the class G(p,p,C), ≤p <p <

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(G(P()))O|)dxdirectly, we could valuate its upperbound by (.). Now we carry on the process as follows

O

GP()–GP()_Oplog₊GP()–GP()_Odx≤C

O

||p_log

+||dx

=C

O

plog₊dx

=C|O|. (.)

Remark It is well known that uniform domains and John domains are the special

Lϕ_{-averaging domains. Therefore, the result of Theorem . remains valid for uniform} domains and John domains.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZD ﬁnished the proof and the writing work. YX and YW gave ZD some excellent advice on the proof and writing. SD gave ZD lots of help in revising the paper. All authors read and approved the ﬁnal manuscript.

Author details

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

University, Seattle, WA 98122, USA.

Acknowledgements

The authors wish to thank the anonymous referees for their time and thoughtful suggestions.

Received: 19 July 2012 Accepted: 9 November 2012 Published: 27 November 2012

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doi:10.1186/1029-242X-2012-271