R E S E A R C H
Open Access
Sharp maximal function inequalities and
boundedness for commutators related to
generalized fractional singular integral
operators
Guangze Gu
*and Mingjie Cai
*Correspondence: toggz@163.com College of Mathematics and Econometrics, Hunan University, Changsha, 410082, P.R. China
Abstract
In this paper, some sharp maximal function inequalities for the commutators related to certain generalized fractional singular integral operators are proved. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin spaces.
MSC: 42B20; 42B25
Keywords: singular integral operator; commutator; sharp maximal function; Morrey space; Triebel-Lizorkin space;BMO; Lipschitz function
1 Introduction and preliminaries
Letb∈BMO(Rn) andTbe the Calderón-Zygmund singular integral operator. The
com-mutator [b,T] generated bybandT is defined by
[b,T]f(x) =b(x)Tf(x) –T(bf)(x).
By using a classical result of Coifmanet al.(see []), we know that the commutator [b,T] is bounded onLp(Rn) ( <p<∞). Now, as the development of singular integral operators
and their commutators, some new integral operators and their commutators are studied (see [–]). In [, , ], the boundedness for the commutators generated by the singu-lar integral operators and Lipschitz functions on Triebel-Lizorkin andLp(Rn) ( <p<∞)
spaces are obtained. In [], some singular integral operators with general kernel are in-troduced, and the boundedness for the operators and their commutators generated by BMOand Lipschitz functions are obtained (see [, ]). Motivated by these, in this paper, we will prove some sharp maximal function inequalities for the commutator associated with certain generalized fractional singular integral operators and theBMOand Lipschitz functions. As an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space.
First, let us introduce some preliminaries. Throughout this paper,Qwill denote a cube of Rnwith sides parallel to the axes. For any locally integrable functionf, the sharp maximal
function off is defined by
M#(f)(x) =sup
Qx
|Q|
Q
f(y) –fQdy,
and here, and in what follows,fQ=|Q|–
Qf(x)dx. It is well known that (see [, ])
M#(f)(x)≈sup
Qx
inf
c∈C
|Q|
Q
f(y) –cdy.
We say that f belongs to BMO(Rn) if M#(f) belongs to L∞(Rn) and define f BMO =
M#(f)
L∞. It is well known that (see [])
f–fkQBMO≤CkfBMO.
Let
M(f)(x) =sup
Qx
|Q|
Q
f(y)dy.
Forη> , letMη(f)(x) =M(|f|η)/η(x).
For <η<nand ≤r<∞, set
Mη,r(f)(x) =sup Qx
|Q|–rη/n
Q
f(y)rdy /r
.
TheApweight is defined by (see [])
Ap=
w∈LlocRn :sup
Q
|Q|
Q
w(x)dx
|Q|
Q
w(x)–/(p–)dx p–
<∞
,
<p<∞,
and
A=
w∈LplocRn :M(w)(x)≤Cw(x), a.e..
Forβ> andp> , letF˙pβ,∞(Rn) be the homogeneous Triebel-Lizorkin space (see []).
Forβ> , the Lipschitz spaceLipβ(Rn) is the space of functionsf such that
fLipβ= sup
x,y∈Rn
x=y
|f(x) –f(y)| |x–y|β <∞.
In this paper, we will study some singular integral operators as follows (see []).
Definition Fix ≤δ<n. LetTδ:S→Sbe a linear operator such thatTδ is bounded
onL(Rn) and has a kernelK, that is, there exists a locally integrable functionK(x,y) on
Rn×Rn\ {(x,y)∈Rn×Rn:x=y}such that
Tδ(f)(x) =
for every bounded and compactly supported functionf, where forKwe have the following: there is a sequence of positive constant numbers{Ck}such that for anyk≥,
|y–z|<|x–y|
K(x,y) –K(x,z)+K(y,x) –K(z,x) dx≤C,
and
k|z–y|≤|x–y|<k+|z–y|
K(x,y) –K(x,z)+K(y,x) –K(z,x) qdy /q
≤Ck
k|z–y| –n/q+δ, ()
where <q< and /q+ /q= . We writeTδ∈GSIO(δ).
Letbbe a locally integrable function onRn. The commutator related toTδis defined by
Tδb(f)(x) =
Rn
b(x) –b(y) K(x,y)f(y)dy.
Remark (a) Note that the classical Calderón-Zygmund singular integral operator satisfies Definition withCj= –jε,ε> andδ= (see [, ]).
(b) Also note that the fractional integral operator with rough kernel satisfies Definition (see []), that is, for <δ<n, letTδbe the fractional integral operator with rough kernel
defined by (see [])
Tδf(x) =
Rn
(x–y) |x–y|n–δf(y)dy,
whereis homogeneous of degree zero onRn,
Sn–(x)dσ(x) = and∈Lipε(Sn–)
for some <ε≤, that is, there exists a constantM> such that for anyx,y∈Sn–,|(x) – (y)| ≤M|x–y|ε. When≡,T
δis the Riesz potential (fractional integral operator).
Definition Letϕ be a positive, increasing function onR+ and there exists a constant D> such that
ϕ(t)≤Dϕ(t) fort≥.
Letf be a locally integrable function onRn. Set, for ≤η<nand ≤p<n/η,
fLp,η,ϕ= sup
x∈Rn,d>
ϕ(d)–pη/n
Q(x,d)
f(y)pdy /p
,
whereQ(x,d) ={y∈Rn:|x–y|<d}. The generalized fractional Morrey space is defined
by
Lp,η,ϕRn =f∈LlocRn :fLp,η,ϕ<∞.
We writeLp,η,ϕ(Rn) =Lp,ϕ(Rn) ifη= , which is the generalized Morrey space. Ifϕ(d) =
dη, η> , thenLp,ϕ(Rn) =Lp,η(Rn), which is the classical Morrey spaces (see [, ]). If
As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of operator on Morrey spaces (see [, , , , ]).
It is well known that commutators are of great interest in harmonic analysis, and they have been widely studied by many authors (see [, ]). In [], Pérez and Trujillo-Gonzalez prove a sharp estimate for the commutator. The main purpose of this paper is to prove some sharp maximal inequalities for the commutatorTδb. As an application, we obtain
theLp-norm inequality, and for Morrey and Triebel-Lizorkin spaces boundedness for the
commutator.
2 Theorems
We shall prove the following theorems.
Theorem Let there be a sequence{Ck} ∈l, <β< ,q≤s<∞and b∈Lipβ(Rn).
Suppose Tδ is a bounded linear operator from Lp(Rn)to Lr(Rn)for any p,r with <p<n/δ
and/r= /p–δ/n,and that it has a kernel K satisfying().Then there exists a constant C> such that,for any f ∈C∞(Rn)andx˜∈Rn,
M#Tδb(f) (˜x)≤CbLipβ
Mβ+δ,s(f)(˜x) +Mβ,s
Tδ(f) (˜x).
Theorem Let there be a sequence{kβC
k} ∈l, <β< ,q≤s<∞and b∈Lipβ(Rn).
Suppose Tδ is a bounded linear operator from Lp(Rn)to Lr(Rn)for any p,r with <p<n/δ
and/r= /p–δ/n,and that it has a kernel K satisfying().Then there exists a constant C> such that,for any f ∈C∞(Rn)andx˜∈Rn,
sup
Q˜x
inf
c
|Q|+β/n
Q
Tδb(f)(x) –cdx≤CbLipβ
Mδ,s(f)(˜x) +Ms
Tδ(f) (˜x).
Theorem Let there be a sequence {kCk} ∈l, q≤s<∞and b∈BMO(Rn). Suppose
Tδ is a bounded linear operator from Lp(Rn)to Lr(Rn)for any p,r with <p<n/δ and
/r= /p–δ/n,and that it has a kernel K satisfying().Then there exists a constant C> such that,for any f∈C∞(Rn)andx˜∈Rn,
M#Tδb(f) (˜x)≤CbBMO
Mδ,s(f)(˜x) +Ms
Tδ(f) (˜x).
Theorem Let there be a sequence{Ck} ∈l, <β <min(,n–δ), q<p<n/(β+δ),
/r= /p– (β+δ)/n and b∈Lipβ(Rn).Suppose T
δis a bounded linear operator from Lp(Rn)
to Lr(Rn)and has a kernel K satisfying().Then Tb
δ is bounded from Lp(Rn)to Lr(Rn).
Theorem Let there be a sequence{Ck} ∈l, <D< n, <β<min(,n–δ),q<p<
n/(β+δ), /r= /p– (β+δ)/n and b∈Lipβ(Rn).Suppose T
δ is a bounded linear operator
from Lp(Rn)to Lr(Rn)and has a kernel K satisfying().Then Tb
δ is bounded from Lp,
β+δ,ϕ(Rn)
to Lr,ϕ(Rn).
Theorem Let there be a sequence{kβC
k} ∈l, <β< ,q<p<n/δ, /r= /p–δ/n
and b∈Lipβ(Rn).Suppose Tδ is a bounded linear operator from Lp(Rn)to Lr(Rn)and has
a kernel K satisfying().Then Tδbis bounded from Lp(Rn)toF˙ β,∞
Theorem Let there be a sequence {kCk} ∈l, q <p<n/δ, /r= /p–δ/n and b∈
BMO(Rn).Suppose T
δis a bounded linear operator from Lp(Rn)to Lr(Rn)and has a kernel
K satisfying().Then Tδbis bounded from Lp(Rn)to Lr(Rn).
Theorem Let there be a sequence{kCk} ∈l, <D< n,q<p<n/δ, /r= /p–δ/n and
b∈BMO(Rn).Suppose T
δ is a bounded linear operator from Lp(Rn)to Lr(Rn)and has a
kernel K satisfying().Then Tδbis bounded from Lp,δ,ϕ(Rn)to Lr,ϕ(Rn).
Corollary Let Tδ∈GSIO(δ),that is,Tδ is the singular integral operator as Definition.
Then Theorems-hold for Tδb.
3 Proofs of theorems
To prove the theorems, we need the following lemma.
Lemma (see []) Let Tδ be the singular integral operator as Definition.Then Tδ is
bounded from Lp(Rn)to Lr(Rn)for <p<n/δand/r= /p–δ/n.
Lemma (see []) For <β< and <p<∞,we have
fF˙β,∞
p ≈
sup
Q·
|Q|+β/n
Q
f(x) –fQdx
Lp≈ sup
Q·infc
|Q|+β/n
Q
f(x) –cdx
Lp .
Lemma (see []) Let <p<∞and w∈≤r<∞Ar.Then,for any smooth function f for
which the left-hand side is finite,
RnM(f)(x)
pw(x)dx≤C
RnM
#
(f)(x)pw(x)dx.
Lemma (see [, ]) Suppose that <η<n, <s<p<n/ηand/r= /p–η/n.Then
Mη,s(f)Lr≤CfLp.
Lemma Let <p<∞, <D< n.Then,for any smooth function f for which the left-hand
side is finite,
M(f)Lp,ϕ≤CM #
(f)Lp,ϕ.
Proof For any cubeQ=Q(x,d) inRn, we knowM(χQ)∈Afor any cubeQ=Q(x,d) by
[]. Noticing thatM(χQ)≤ andM(χQ)(x)≤dn/(|x–x|–d)nifx∈Qc, by Lemma , we
have, forf ∈Lp,ϕ(Rn),
Q
M(f)(x)pdx=
RnM(f)(x)
pχ Q(x)dx
≤
Rn
M(f)(x)pM(χQ)(x)dx
≤C
RnM
#
=C
Q
M#(f)(x)pM(χQ)(x)dx
+
∞
k=
k+Q\kQM
#
(f)(x)pM(χQ)(x)dx
≤C
Q
M#(f)(x)pdx+
∞
k=
k+Q\kQM
#
(f)(x)p |Q| |k+Q|dx
≤C
Q
M#(f)(x)pdx+
∞
k=
k+Q
M#(f)(x)p–kndy
≤CM#(f)pLp,ϕ ∞
k=
–knϕk+d ≤CM#(f)pLp,ϕ ∞
k=
–nDkϕ(d) ≤CM#(f)pLp,ϕϕ(d),
thus
ϕ(d)
Q
M(f)(x)pdx /p
≤C
ϕ(d)
Q
M#(f)(x)pdx /p
and
M(f)Lp,ϕ≤CM #
(f)Lp,ϕ.
This finishes the proof.
Lemma Let Tδ be a bounded linear operator from Lp(Rn)to Lr(Rn)for any p, r with
<p<n/δand/r= /p–δ/n, <D< n.Then
Tδ(f)Lr,ϕ≤CfLp,δ,ϕ.
Lemma Let <D< n, ≤s<p<n/ηand/r= /p–η/n.Then
Mη,s(f)Lr,ϕ≤CfLp,η,ϕ.
The proofs of two Lemmas are similar to that of Lemma by Lemmas and , we omit the details.
Proof of Theorem It suffices to prove forf ∈C∞(Rn) and some constantC that the
following inequality holds:
|Q|
Q
Tδb(f)(x) –Cdx≤CbLipβ
Mβ+δ,s(f)(˜x) +Mβ,s
Tδ(f) (˜x).
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c,
Tδb(f)(x) =
b(x) –bQ Tδ(f)(x) –Tδ
(b–bQ)f (x) –Tδ
Then
|Q|
Q
Tb
δ(f)(x) –Tδ
(bQ–b)f (x)dx
≤
|Q|
Q
b(x) –bQ Tδ(f)(x)dx+
|Q|
Q
Tδ
(b–bQ)f (x)dx
+
|Q|
Q
Tδ
(b–bQ)f (x) –Tδ
(b–bQ)f (x)dx
=I+I+I.
ForI, by Hölder’s inequality, we obtain
I≤
|Q|xsup∈Q
b(x) –bQ
Q
Tδ(f)(x)dx
≤
|Q|xsup∈Q
b(x) –bQ|Q|–/s
Q
Tδ(f)(x)sdx
/s
≤CbLipβ|Q|–/s|Q|β/n|Q|/s–β/n
|Q|–sβ/n
Q
Tδ(f)(x)
s
dx /s
≤CbLipβMβ,s
Tδ(f) (˜x).
ForI, choose <r<∞such that /r= /s–δ/n, by (Ls,Lr)-boundedness ofTδ, we get
I≤
|Q|
Rn Tδ
(b–bQ)f (x)
r
dx /r
≤C|Q|–/r
Rn
b(x) –bQ f(x)sdx
/s
≤C|Q|–/rbLipβ|Q|β/n|Q|/s–(β+δ)/n
|Q|–s(β+δ)/n
Q
f(x)sdx /s
≤CbLipβMβ+δ,s(f)(˜x).
ForI, recalling thats>q, we have
I≤
|Q|
Q
(Q)c
b(y) –bQf(y)K(x,y) –K(x,y)dy dx
≤| Q| Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)b(y) –bk+Qf(y)dy dx
+
|Q| Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)bk+Q–bQf(y)dy dx
≤|C
Q| Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)
q
dy /q
× sup
y∈k+Q
b(y) –bk+Q
k+Q
f(y)qdy /q
+ C |Q|
Q
∞
k=
|bk+Q–bQ|
kd≤|y–x |<k+d
K(x,y) –K(x,y)
q
dy /q
×
k+Q
f(y)qdy /q dx ≤C ∞ k= Ck
kd –n/q+δk+Qβ/nbLipβk+Q/q –/s
k+Q/s–(β+δ)/n
×
|k+Q|–s(β+δ)/n
k+Q
f(y)sdy /s
+C
∞
k=
bLipβkQ
β/n
Ck
kd –n/q+δk+Q/q–/sk+Q/s–(β+δ)/n
×
|k+Q|–s(β+δ)/n
k+Q
f(y)sdy /s
≤CbLipβMβ+δ,s(f)(˜x)
∞
k=
Ck≤CbLipβMβ+δ,s(f)(˜x).
These complete the proof of Theorem .
Proof of Theorem It suffices to prove forf ∈C∞(Rn) and some constantC
that the
following inequality holds:
|Q|+β/n
Q
Tδb(f)(x) –Cdx≤CbLipβ
Mδ,s(f)(x˜) +Ms
Tδ(f)(x˜) .
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c, Tδb(f)(x) =
b(x) –bQ Tδ(f)(x) –Tδ
(b–bQ)f (x) –Tδ
(b–bQ)f (x).
Then
|Q|+β/n
Q
Tδb(f)(x) –Tδ
(bQ–b)f (x)dx
≤|
Q|+β/n
Q
b(x) –bQ Tδ(f)(x)dx+
|Q|+β/n
Q
Tδ
(b–bQ)f (x)dx
+
|Q|+β/n
Q
Tδ
(b–bQ)f (x) –Tδ
(b–bQ)f (x)dx
=II+II+II.
By using the same argument as in the proof of Theorem , we get, for <r<∞ with /r= /s–δ/n,
II≤ C |Q|+β/n sup
x∈Q
b(x) –bQ|Q|–/s
Q
Tδ(f)(x)
s
dx /s
≤C|Q|––β/nb
Lipβ|Q|β/n|Q|–/s|Q|/s
|Q|
Q
Tδ(f)(x)
s
dx /s
≤CbLipβMs
II≤
|Q|+β/n|Q|
–/r
Rn Tδ
(b–bQ)f (x)rdx
/r
≤| C
Q|+β/n|Q|
–/r
Rn
b(x) –bQ f(x)
s
dx /s
≤| C
Q|+β/n|Q|
–/rb
Lipβ|Q|β/n|Q|/s–δ/n
|Q|–sδ/n
Q
f(x)sdx /s
≤CbLipβMδ,s(f)(x˜),
II≤
|Q|+β/n
Q
(Q)c
b(y) –bQf(y)K(x,y) –K(x,y)dy dx
≤
|Q|+β/n
Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)b(y) –bk+Qf(y)dy dx
+
|Q|+β/n
Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)bk+Q–bQf(y)dy dx
≤ C
|Q|+β/n
Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)
q
dy /q
× sup
y∈k+Q
b(y) –bk+Q
k+Q
f(y)qdy /q
dx
+ C
|Q|+β/n
Q
∞
k=
|bk+Q–bQ|
kd≤|y–x |<k+d
K(x,y) –K(x,y)
q
dy /q
×
k+Q
f(y)qdy /q
dx
≤C|Q|–β/n
∞
k= Ck
kd –n/q+δk+Qβ/nbLipβk+Q
/q–δ/n
×
|k+Q|–sδ/n
k+Q
f(y)sdy /s
+C|Q|–β/n
∞
k=
bLipβkQ
β/n
Ck
kd –n/q+δk+Q/q–δ/n
×
|k+Q|–sδ/n
k+Q
f(x)sdx /s
≤CbLipβMδ,s(f)(˜x)
∞
k=
kβCk
≤CbLipβMδ,s(f)(˜x).
This completes the proof of Theorem .
Proof of Theorem It suffices to prove forf ∈C∞(Rn) and some constantC
that the
following inequality holds:
|Q|
Q
Tδb(f)(x) –Cdx≤CbBMO
Mδ,s(f)(x˜) +Ms
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c, Tδb(f)(x) =b(x) –bQ Tδ(f)(x) –Tδ
(b–bQ)f (x) –Tδ
(b–bQ)f (x).
Then
|Q|
Q
Tδb(f)(x) –Tδ
(bQ–b)f (x)dx
≤
|Q|
Q
b(x) –bQ Tδ(f)(x)dx+
|Q|
Q
Tδ
(b–bQ)f (x)dx
+
|Q|
Q
Tδ
(b–bQ)f (x) –Tδ
(b–bQ)f (x)dx
=III+III+III.
ForIII, by Hölder’s inequality, we get
III≤
|Q|
Q
b(x) –bQ s
dx /s
|Q|
Q
Tδ(f)(x)
s
dx /s
≤CbBMOMs
Tδ(f)(˜x).
ForIII, choose <p<ssuch that /r= /p–δ/n, by Hölder’s inequality and (Lp,Lr
)-boundedness ofTδ, we obtain
III≤
|Q|
Rn Tδ
(b–bQ)f (x)rdx
/r
≤C|Q|–/r
Rn
(b–bQ)f(x)pdx
/p
≤C|Q|–/r
Q
b(x) –bQsp/(s–p)dx
(s–p)/sp
Q
f(x)sdx /s
≤C|Q|–/r|Q|(s–p)/sp
|Q|
Q
b(x) –bQsp/(s–p)dx
(s–p)/sp
× |Q|/s–δ/n
|Q|–sδ/n
Q
f(x)sdx /s
≤CbBMO|Q|–/r+/p–δ/n
|Q|–sδ/n
Q
f(x)sdx /s
≤CbBMOMδ,s(f)(x˜).
ForIII, recalling thats>q, taking <p<∞with /p+ /q+ /s= , by Hölder’s inequality,
we obtain
III≤
|Q|
Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)b(y) –bQf(y)dy dx
≤
|Q| Q ∞ k=
kd≤|y–x |<k+d
K(x,y) –K(x,y)qdy
/q
×
k+Q
b(x) –bQ p
dx /p
k+Q
f(y)sdy /s
≤C
∞
k= Ck
kd –n/q+δkkd n/pbBMO
kd n(/s–δ/n)
×
|k+Q|–sδ/n
k+Q
f(y)sdy /s
≤CbBMO
∞
k= kCk
kd n(–+/q+/p+/s)
|k+Q|–sδ/n
k+Q
f(y)sdy /s
≤CbBMOMδ,s(f)(x˜)
∞
k= kCk
≤CbBMOMδ,s(f)(˜x).
This completes the proof of Theorem .
Proof of Theorem Chooseq<s<pin Theorem and let /t= /p–δ/n, then /r= /t–β/n, thus we have, by Lemmas , , and ,
Tδb(f)Lr ≤M
Tδb(f) Lr≤CM#
Tδb(f) Lr ≤CbLipβMβ,s
Tδ(f) Lr+Mβ+δ,s(f)Lr ≤CbLipβTδ(f)Lt+fLp
≤CbLipβfLp.
This completes the proof of Theorem .
Proof of Theorem Chooseq<s<pin Theorem and let /t= /p–δ/n, then /r= /t–β/n, thus we have, by Lemmas -,
Tδb(f)Lr,ϕ≤M
Tδb(f) Lr,ϕ≤CM #
Tδb(f) Lr,ϕ
≤CbLipβMβ,s
Tδ(f) Lr,ϕ+Mβ+δ,s(f)Lr,ϕ
≤CbLipβTδ(f)Lt,β,ϕ+fLp,β+δ,ϕ
≤CbLipβfLp,β+δ,ϕ.
This completes the proof of Theorem .
Proof Theorem Chooseq<s<pin Theorem . By using Lemma , we obtain
Tδb(f)F˙rβ,∞≤C sup
Q·
|Q|+β/n
Q
Tδb(f)(x) –Tδ
(bQ–b)f (x)dx
Lr ≤CbLipβMs
Tδ(f) Lr+Mδ,s(f)Lr ≤CbLipβTδ(f)Lr+fLp
≤CbLipβfLp.
Proof of Theorem Chooseq≤s<pin Theorem and similar to the proof of Theorem , we have
Tδb(f)Lr ≤M
Tδb(f) Lr≤CM#
Tδb(f) Lr
≤CbBMOMs
Tδ(f) Lr+Mδ,s(f)Lr ≤CbBMOTδ(f)Lr+fLp
≤CbBMOfLp.
This completes the proof of the theorem.
Proof of Theorem Chooseq≤s<pin Theorem and similar to the proof of Theorem , we have
Tδb(f)Lr,ϕ≤M
Tδb(f) Lr,ϕ≤CM #
Tδb(f) Lr,ϕ
≤CbBMOMs
Tδ(f) Lr,ϕ+Mδ,s(f)Lr,ϕ
≤CbBMOTδ(f)Lr,ϕ+fLp,δ,ϕ
≤CbBMOfLp,δ,ϕ.
This completes the proof of the theorem.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper together. They also read and approved the final manuscript.
Acknowledgements
Project supported by Hunan Provincial Natural Science Foundation of China (12JJ6003) and Scientific Research Fund of Hunan Provincial Education Departments (12K017).
Received: 9 March 2014 Accepted: 25 April 2014 Published:23 May 2014
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10.1186/1029-242X-2014-211
