Weighted boundedness of multilinear singular integral operators

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R E S E A R C H

Open Access

Weighted boundedness of multilinear

singular integral operators

Wei-Ping Kuang

1

_{and Zhi-Gang Wang}

2*

*_{Correspondence:}

[email protected] 2_{School of Mathematics and}

Statistics, Anyang Normal University, Anyang, Henan 455000, People’s Republic of China

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

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear singular integral operators. As an application, we obtain the boundedness of the multilinear operators on weighted Lebesgue and Morrey spaces.

MSC: 42B20; 42B25

Keywords: multilinear operator; singular integral operator; sharp maximal function;

weighted BMO; weighted Lipschitz function

1 Introduction

As the development of singular integral operators (see [–]), their commutators opera-tors have been well studied. In [–], the authors prove that the commutaopera-tors generated by the singular integral operators andBMOfunctions are bounded onLp₍_Rn_{) for  <}_p_<_∞_. Chanillo (see []) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [, ], the boundedness for the commutators gener-ated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and

Lp(Rn) ( <p<∞) spaces are obtained. In [, ], the boundedness for the commutators generated by the singular integral operators and the weightedBMOand Lipschitz func-tions onLp₍_Rn_{) ( <}_p_<_∞_{) spaces are obtained (also see [, ]). In [–], the authors} studied some multilinear singular integral operators as follows (also see [, ]):

Tb(f)(x) =

Rm+(b;x,y)

|x–y|m K(x,y)f(y)dy,

and they obtained some variant sharp function estimates and boundedness of the multi-linear operators ifDα_b_∈_BMO(_Rn_{) for all}_α_with_|_α_|₌_m_{. In this paper, we will study the} multilinear operator generated by the singular integral operator and the weighted Lips-chitz andBMOfunctions, that is,Dα_b_∈_BMO(_w_{) or}_Dα_b_∈_Lip

β(w) for allαwith|α|=m.

2 Preliminaries

First, let us introduce some notations. Throughout this paper,Qwill denote a cube ofRn with 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;

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here, and in the following,fQ=|Q|–

Qf(x)dx. It is well known that (see [, ])

M#(f)(x)≈sup Qx

inf c∈C



|Q|

Q

f(y) –cdy.

Let

M(f)(x) =sup Qx



|Q|

Q

f(y)dy.

Forη> , letM#

η(f)(x) =M#(|f|η)/η(x) andMη(f)(x) =M(|f|η)/η(x).

For  <η<n, ≤p<∞and the non-negative weight functionw, set

Mη,p,w(f)(x) =sup Qx



w(Q)–pη/n

Q

f(y)pw(y)dy /p

.

We writeMη,p,w(f) =Mp,w(f) ifη= .

TheApweight is deﬁned by (see []), for  <p<∞,

Ap=

w∈L_locRn :sup Q



|Q|

Q

w(x)dx



|Q|

Q

w(x)–/(p–)dx p–

<∞

and

A=

w∈Lp_locRn :M(w)(x)≤Cw(x), a.e..

Given a non-negative weight functionw. For ≤p<∞, the weighted Lebesgue space

Lp(Rn,w) is the space of functionsf such that

fLp(w)=

Rn

f(x)pw(x)dx /p

<∞.

For  <β<  and the non-negative weight functionw, the weighted Lipschitz space Lip_β(w) is the space of functionsbsuch that

bLip_β(w)=sup Q



w(Q)β/n



w(Q)

Q

b(y) –bQpw(x)–pdy

/p <∞,

and the weightedBMOspaceBMO(w) is the space of functionsbsuch that

bBMO(w)=sup Q



w(Q)

Q

b(y) –bQ p

w(x)–pdy /p

<∞.

Remark

() It is well known that (see [, ]), forb∈Lip_β(w),w∈Aandx∈Q,

|bQ–bk_Q| ≤CkbLip_β(w)w(x)w

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() It is well known that (see [, ]), forb∈BMO(w),w∈Aandx∈Q,

|bQ–bk_Q| ≤CkbBMO(w)w(x).

() Letb∈Lip_β(w)orb∈BMO(w)andw∈A. By [], we know that spacesLipβ(w) orBMO(w)coincide and the normsbLip_β(w)orbBMO(w)are equivalent with

respect to diﬀerent values≤p<∞.

Deﬁnition  Letϕbe a positive, increasing function onR+and let there exist a constant

D>  such that

ϕ(t)≤Dϕ(t) fort≥.

Letwbe a non-negative weight function onRn_and_f_{be a locally integrable function on}_Rn_. Set, for ≤p<∞,

fLp,ϕ_(w)= sup x∈Rn_,d>

 ϕ(d)

Q(x,d)

f(y)pw(y)dy /p

,

whereQ(x,d) ={y∈Rn_:_|_x_–_y_|_<_d_}_{. The generalized weighted Morrey space is deﬁned by}

Lp,ϕRn,w =f ∈L_locRn :fLp,ϕ_(w)<∞.

Ifϕ(d) =dδ_,_δ_{> , then}_Lp,ϕ₍_Rn_,_w_{) =}_Lp,δ₍_Rn_,_w_{), which is the classical Morrey spaces (see} [, ]). Ifϕ(d) = , thenLp,ϕ₍_Rn_,_w_{) =}_Lp₍_Rn_,_w_{), which is the weighted Lebesgue spaces} (see []).

As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [, –]).

In this paper, we will study the singular integral operators as follows (see []).

Deﬁnition  LetT:S→S be a linear operator such thatT is bounded onLp₍_Rn_{) for}  <p<∞and weak (L_,_L_{)-bounded and there exists a locally integrable function}_K₍_x_,_y₎ onRn×Rn\ {(x,y)∈Rn×Rn:x=y}such that

T(f)(x) =

RnK(x,y)f(y)dy

for every bounded and compactly supported functionf, whereKsatisﬁes, for ﬁxedε> ,

K(x,y)≤C|x–y|–n

and

K(y,x) –K(z,x)+K(x,y) –K(x,z)≤C|y–z|ε|x–z|–n–ε

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Moreover, letmbe the positive integer andbbe the function onRn_{. Set}

Rm+(b;x,y) =b(x) –

|α|≤m  α!D

α_b₍_y₎₍_x_–_y₎α_.

The multilinear operator related to the operatorTis deﬁned by

Tb(f)(x) =

Rn

Rm+(b;x,y)

|x–y|m K(x,y)f(y)dy.

Note that the classical Calderón-Zygmund singular integral operator satisﬁes the con-ditions of Deﬁnition  (see [, ]) and that the commutator [b,T](f) =bT(f) –T(bf) is a particular operator of the multilinear operatorTbifm= . The multilinear operatorTb

are the non-trivial generalizations of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [, , ]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operatorTb_{. As the application, we obtain} the weightedLp_{-norm inequality and Morrey space boundedness for the multilinear} op-eratorTb_.

3 Theorems and lemmas

We shall prove the following theorems.

Theorem  Let T be the singular integral operator as Deﬁnition,w∈A,  <η< ,  <

r<∞and Dα_b_∈_BMO(_w₎_{for all}_α_with_|_α_|₌_m_._{Then there exists a constant C}_{> }_such that,for any f∈C_∞(Rn₎_and_x_˜_∈_Rn_,

M#η

Tb(f) (x˜)≤C |α|=m

Dαb_BMO_(w)w(x˜)Mr,w(f)(x˜).

Theorem  Let T be the singular integral operator as Deﬁnition,w∈A,  <η< ,  <

r<∞,  <β< and Dα_b_∈_Lip

β(w)for allα with|α|=m.Then there exists a constant C> such that,for any f ∈C∞_(Rn₎_and_x_˜_∈_Rn_,

M#η

Tb(f) (x˜)≤C |α|=m

_Dα b_Lip

β(w)w(x˜)Mβ,r,w(f)(x˜).

Theorem  Let T be the singular integral operator as Deﬁnition,w∈A,  <p<∞and

Dα_b_∈_BMO(_w₎_{for all}_α_with_|_α_|₌_m_._{Then T}b_{is bounded from L}p₍_Rn_,_w₎_{to L}p₍_Rn_,_w–p_).

Theorem  Let T be the singular integral operator as Deﬁnition ,w∈A,  <p<∞,  <D< n_{and D}α_b_∈_BMO(_w₎_{for all}_α_with_|_α_|₌_m_._{Then T}b_{is bounded from L}p,ϕ₍_Rn_,_w₎

to Lp,ϕ₍_Rn_,_w–p_).

Theorem  Let T be the singular integral operator as Deﬁnition,w∈A,  <β< ,  <

p<n/β, /q= /p–β/n and Dα_b_∈_Lip

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Theorem  Let T be the singular integral operator as Deﬁnition,w∈A,  <β< ,  <

D< n,  <p<n/β, /q= /p–β/n and Dα_b_∈_Lip

β(w)for allα with|α|=m.Then Tbis bounded from Lp,ϕ₍_Rn_,_w₎_{to L}q,ϕ₍_Rn_,_w–q_).

To prove the theorems, we need the following lemmas.

Lemma (See [, p.]) Let <p<q<∞and for any function f ≥.We deﬁne,for

/r= /p– /q,

fWLq=sup

λ>

λx∈Rn:f(x) >λ/q, Np,q(f) =sup

Q f

χQLp/χ_Q_Lr,

where the sup is taken for all measurable sets Q with <|Q|<∞.Then

fWLq≤N_p,q(f)≤q/(q–p) /pf_WLq.

Lemma (See [, ]) Let≤η<n, ≤s<p<n/η, /q= /p–η/n and w∈A.Then

_M_η_,s,w(f)_Lq_(w)≤CfLp_(w).

Lemma (See []) Let <p,η<∞and w∈__≤_r<_∞Ar.Then,for any smooth function f

for which the left-hand side is ﬁnite,

Rn

Mη(f)(x)pw(x)dx≤C

Rn

M#_η(f)(x)pw(x)dx.

Lemma (See []) Let <p<∞,  <η<∞,  <D< n_{and w}_∈_A

.Then,for any smooth

function f for which the left-hand side is ﬁnite,

Mη(f)_Lp,ϕ_(w)≤CM #

η(f)Lp,ϕ_(w).

Lemma (See []) Let≤η<n,  <D< n_{, }_≤_s_<_p_<_n_/_η_{, /}_q_{= /}_p_–_η_/_{n and w}_∈_A .

Then

_M_η_,s,w(f)_Lq,ϕ_(w)≤CfLp,ϕ_(w).

Lemma (See []) Let b be a function on Rnand Dα_A_∈_Lq₍_Rn₎_{for all}_α_with_|_α_|₌_{m and}

any q>n.Then

Rm(b;x,y)≤C|x–y|m

|α|=m



| ˜Q(x,y)|

˜

Q(x,y)

Dαb(z)qdz /q

,

whereQ is the cube centered at x and having side length˜ √n|x–y|.

4 Proofs of theorems

Proof of Theorem It suﬃces to prove forf ∈C_∞(Rn_{) and some constant}_C

, the following inequality holds:



|Q|

Q

Tb(f)(x) –C

η dx

/η

≤C |α|=m

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Fix a cubeQ=Q(x,d) andx˜∈Q. LetQ˜ = √nQandb˜(x) =b(x) –

|α|=mα!(D

α_b₎ ˜

Qxα, thenRm(b;x,y) =Rm(b˜;x,y) andDαb˜=Dαb– (Dαb)Q˜ for|α|=m. We write, forf=fχQ˜ andf=fχRn_{\ ˜}_Q,

Tb(f)(x) =

Rn

Rm(b˜;x,y)

|x–y|m K(x,y)f(y)dy–

|α|=m  α!

Rn

(x–y)α_Dα_b˜₍_y₎

|x–y|m K(x,y)f(y)dy

+

Rn

Rm+(b˜;x,y)

|x–y|m K(x,y)f(y)dy

=T

Rm(b˜;x,·)

|x–·|m f

–T |α|=m

 α!

(x–·)α_Dα_b˜ |x–·|m f

+Tb˜(f)(x),

then



|Q|

Q

Tb(f)(x) –Tb(f)(x)

η dx

/η

≤C



|Q|

Q

T

Rm(b˜;x,·)

|x–·|m f

ηdx

/η

+C



|Q|

Q

T

|α|=m

(x–·)α_Dα_b˜ |x–·|m f

ηdx

/η

+C



|Q|

Q

T˜b(f)(x) –Tb˜(f)(x)

η dx

/η

=I+I+I.

ForI, noting thatw∈A,wsatisﬁes the reverse of Hölder’s inequality:



|Q|

Q

w(x)p_dx

/p

≤_|Q|C

Q

w(x)dx

for all cubeQand some  <p<∞(see []). We takeq=rp/(r+p– ) in Lemma  and have  <q<randp=q(r– )/(r–q), then by the Lemma  and Hölder’s inequality, we obtain

Rm(b˜;x,y)

≤C|x–y|m |α|=m



| ˜Q(x,y)|

˜

Q(x,y)

Dα_b˜₍_z₎q_dz /q

≤C|x–y|m

|α|=m

| ˜Q|–/q

˜

Q(x,y)

Dαb˜(z)qw(z)q(–r)/rw(z)q(r–)/rdz /q

≤C|x–y|m

|α|=m

| ˜Q|–/q

˜

Q(x,y)

Dαb˜(z)rw(z)–rdz /r

×

˜

Q(x,y)

w(z)q(r–)/(r–q)dz

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≤C|x–y|m

|α|=m

| ˜Q|–/q_Dα

b_BMO_(w)w(Q˜)/r| ˜Q|(r–q)/rq

×



| ˜Q(x,y)|

˜

Q(x,y)

w(z)p_dz

(r–q)/rq

≤C|x–y|m |α|=m

Dαb_BMO_(w)| ˜Q|–/qw(Q˜)/r| ˜Q|/q–/r



| ˜Q(x,y)|

˜

Q(x,y)

w(z)dz (r–)/r

≤C|x–y|m |α|=m

Dα_b

BMO(w)| ˜Q|

–/q_w₍_Q˜₎/r_{| ˜}_Q_|/q–/r_w₍_Q˜₎–/r_{| ˜}_Q_|/r–

≤C|x–y|m

|α|=m

Dαb_BMO_(w)w(Q˜) | ˜Q|

≤C|x–y|m

|α|=m

Dαb_BMO_(w)w(x˜),

thus, by theLs_{-boundedness of}_T_{(see Lemma ) for  <}_s_<_r_and_w_∈_A

⊆Ar/s, we obtain

I≤

C |Q|

Q

T

Rm(b˜;x,·)

|x–·|m f

dx

≤C |α|=m

Dαb_BMO_(w)w(x˜)



|Q|

Rn

T(f)(x) s

dx /s

≤C |α|=m

Dαb_BMO_(w)w(x˜)|Q|–/s

Rn

f(x) s

dx /s

≤C |α|=m

˜

Q

f(x)sw(x)s/rw(x)–s/rdx /s

≤C |α|=m

˜

Q

f(x)rw(x)dx /r

˜

Q

w(x)–s/(r–s)dx (r–s)/rs

≤C |α|=m

Dαb_BMO_(w)w(x˜)|Q|–/sw(Q˜)/r



w(Q˜)

˜

Q

f(x)rw(x)dx /r

×



| ˜Q|

˜

Q

w(x)–s/(r–s)dx

(r–s)/rs 

| ˜Q|

˜

Q

w(x)dx /r

| ˜Q|/s_w₍_Q˜₎–/r

≤C |α|=m

ForI, by the weak (L,L) boundedness ofT (see Lemma ) and Kolmogoro’s inequality (see Lemma ), we obtain

I≤C

|α|=m



|Q|

Q

TDαbf˜ (x)

η dx

/η

≤C |α|=m

|Q|/η–

|Q|/η

T(Dα_bf˜

)χQLη

χQLη/(–η)

≤C |α|=m



|Q|T

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≤C |α|=m



|Q|

Rn

Dαb˜(x)f(x)dx

≤C |α|=m



|Q|

˜

Q

Dα_b₍_x_{) –}_Dα_b ˜

Qw(x)

–/r_f₍_x₎_w₍_x₎/r_dx

≤C |α|=m



|Q|

˜

Q

_Dα

b(x) –Dαb _Q_˜ rw(x)–rdx /r

˜

Q

_f₍_x₎r

w(x)dx /r

≤C |α|=m



|Q|DαbBMO(w)w(Q˜)

/r_w₍_Q˜₎/r



w(Q˜)

˜

Q

f(x)rw(x)dx /r

≤C |α|=m

Dαb_BMO_(w)w(Q˜)

| ˜Q| Mr,w(f)(x˜)

≤C |α|=m

_Dα

b_BMO_(w)w(x˜)Mr,w(f)(x˜).

ForI, note that|x–y| ≈ |x–y|forx∈Qandy∈Rn\Q, we write

Tb˜(f)(x) –Tb˜(f)(x)

=

Rn

Rm(b˜;x,y) –Rm(b˜;x,y)

K(x,y)

|x–y|mf(y)dy

+

Rn

K(x,y)

|x–y|m –

K(x,y)

|x–y|m

Rm(b˜;x,y)f(y)dy

+

|α|=m  α!

Rn

K(x,y)

|x–y|m –

K(x,y)

|x–y|m

(x–y)α_Dα_b˜₍_y₎_f

(y)dy

+

|α|=m  α!

Rn

(x–y)α |x–y|m –

(x–y)α

|x–y|m

K(x,y)Dαb˜(y)f(y)dy

=I_()(x) +I_()(x) +I_()(x) +I_()(x).

ForI_()(x), by the formula (see []):

Rm(b˜;x,y) –Rm(b˜;x,y) =

|γ|<m  γ!Rm–|γ|

Dγb˜;x,x (x–y)γ

and Lemma , we have, similar to the proof ofI,

Rm(b˜;x,y) –Rm(b˜;x,y)≤C

|γ|<m

|α|=m

|x–x|m–|γ||x–y||γ|Dαb_BMO_(w)w(x˜),

thus, byw∈A⊆Ar,

_I()  (x)

≤ ∞

k=

k+_Q_˜_\_k_Q˜

Rm(b˜;x,y) –Rm(b˜;x,y)|K (x,y)|

|x–y|mf(y)dy

≤C |α|=m

∞

k=

k+_Q_˜_\_k_Q_˜

|x–x|

|x–y|n+

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≤C |α|=m

∞

k=

d

(k_d₎n+

k_Q˜

f(y)w(y)/rw(y)–/rdy

≤C |α|=m

× ∞

k=

d

(k_d₎n+

k_Q_˜

f(y)rw(y)dy /r

k_Q_˜ w(y)

–/(r–)_dy

(r–)/r

≤C |α|=m

∞

k=

d

(k_d₎n+w

kQ˜ /r



w(k_Q˜₎

k_Q_˜

f(y)rw(y)dx /r

×



|k_Q˜_|

k_Q_˜w(y)

–/(r–)_dy

(r–)/r 

|k_Q˜_|

k_Q_˜w(y)dy

/r

kQ˜wkQ˜ –/r

≤C |α|=m

_Dα

b_BMO_(w)w(x˜)Mr,w(f)(x˜)

∞

k= –k

≤C |α|=m

Dα_b

BMO(w)w(x˜)Mr,w(f)(x˜).

ForI_()(x), by the conditions ofK, we get

I_()(x)≤C ∞

k=

k+_Q_˜_\_k_Q_˜

_|K_x(_–x,_yy_|m) –

K(x,y)

|x–y|m

Rm(b˜;x,y)f(y)dy

≤C |α|=m

∞

k=

k+_Q_˜_\_k_Q_˜

_|x

–x|

|x–y|n+

+ |x–x|

ε

|x–y|n+ε

f(y)dy

≤C |α|=m

Dα_b

BMO(w)w(x˜)

∞

k=

d

(k_d₎n++

dε

(k_d₎n+ε

k_Q˜

f(y)dy

≤C |α|=m

Dαb_BMO_(w)w(x˜)Mr,w(f)(x˜)

∞

k=

–k+ –kε

≤C |α|=m

Similarly, we have

I_()(x)+I_()(x)

≤C |α|=m

∞

k=

k+_Q_˜_\_k_Q_˜

d

(k_d₎n+ +

dε

(k_d₎n+ε

f(y)Dαb˜(y)dy

≤C |α|=m

∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

×

k_Q˜

Dαb(y) –Dαb _k_Q˜w(y)–/rf(y)w(y)/rdy

+C

|α|=m

∞

k=

d

(k_d₎n+ +

dε

(10)

×

k_Q˜

Dαb _k_Q˜ –

Dαb _Q_˜f(y)w(y)/rw(y)–/rdy

≤C |α|=m

∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

k_Q_˜

Dαb(y) –Dαb _k_Q˜ r

w(y)–rdy /r

×

k_Q˜

f(y)rw(y)dy /r

+C

|α|=m

∞

k=

k

d

(k_d₎n+ +

dε

(k_d₎n+ε

k_Q_˜

f(y)rw(y)dx /r

×



|k_Q˜_|

k_Q_˜w(y)

–/(r–)_dy

(r–)/r 

|k_Q˜_|

k_Q_˜w(y)dy

/r

≤C |α|=m

_Dα

b_BMO_(w) ∞

k=

–k_{+ }–kε w(kQ˜) |kQ|˜



w(k_Q˜₎

k_Q˜

_f(y)rw(y)dx /r

+C

|α|=m

∞

k=

k–k+ –kε 

w(k_Q˜₎

k_Q_˜

f(y)rw(y)dx /r

≤C |α|=m

Thus

I≤C

|α|=m

These complete the proof of Theorem .

Proof of Theorem It suﬃces to prove forf ∈C∞_ (Rn_{) and some constant}_C

, the following inequality holds:



|Q|

Q

Tb(f)(x) –C

η dx

/η

≤C |α|=m

Dαb_Lip

Fix a cubeQ=Q(x,d) andx˜∈Q. Similar to the proof of Theorem , we have, forf=fχQ˜ andf=fχRn_{\ ˜}_Q,



|Q|

Q

Tb(f)(x) –Tb˜(f)(x)

η dx

/η

≤C



|Q|

Q

T

Rm(b˜;x,·)

|x–·|m f

ηdx

/η

+C



|Q|

Q

T

|α|=m

(x–·)α_Dα_b˜ |x–·|m f

ηdx

/η

+C



|Q|

Q

T˜b₍_f

)(x) –Tb˜(f)(x)

η dx

/η

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ForJandJ, by using the same argument as in the proof of Theorem , we get

Rm(b˜;x,y)

≤C|x–y|m

|α|=m

| ˜Q|–/q

˜

Q(x,y)

Dαb˜(z)qw(z)q(–r)/rw(z)q(r–)/rdz /q

≤C|x–y|m

×

|α|=m

| ˜Q|–/q

˜

Q(x,y)

Dαb˜(z)rw(z)–rdz /r

˜

Q(x,y)

w(z)q(r–)/(r–q)dz (r–q)/rq

≤C|x–y|m

×

|α|=m

| ˜Q|–/q_Dα b_Lip

β(w)w(Q˜)

β/n+/r_{| ˜}_Q|(r–q)/rq



| ˜Q(x,y)|

˜

Q(x,y)

w(z)p_dz

(r–q)/rq

≤C|x–y|m

×

|α|=m

Dαb_Lip

β(w)| ˜Q|

–/q_w₍_Q˜₎β/n+/r_{| ˜}_Q|/q–/r



| ˜Q(x,y)|

˜

Q(x,y)

w(z)dz (r–)/r

≤C|x–y|m |α|=m

Dα_b

Lip_β(w)| ˜Q|

–/q_w₍_Q˜₎β/n+/r_{| ˜}_Q_|/q–/r_w₍_Q˜₎–/r_{| ˜}_Q_|/r–

≤C|x–y|m |α|=m

Dα_b

Lip_β(w)

w(Q˜)β/n+

| ˜Q|

≤C|x–y|m

|α|=m

Dαb_Lip

β(w)w(

˜

Q)β/nw(x˜),

thus

J≤C

|α|=m

Dαb_Lip

β(w)w(Q˜)

β/n_w₍_x_˜₎_|Q|–/s

Rn

f(x) s

dx /s

≤C |α|=m

Dαb_Lip

β(w)w(

˜

Q)β/nw(x˜)|Q|–/s

˜

Q

f(x)rw(x)dx /r

× ˜

Q

w(x)–s/(r–s)dx (r–s)/rs

≤C |α|=m

Dαb_Lip

β(w)w(x˜)| ˜Q|

–/s_w₍_Q˜₎/r



w(Q˜)–rβ/n

˜

Q

f(x)rw(x)dx /r

×



| ˜Q|

˜

Q

w(x)–s/(r–s)dx

(r–s)/rs 

| ˜Q|

˜

Q

w(x)dx /r

| ˜Q|/s_w₍_Q˜₎–/r

≤C |α|=m

_Dα b_Lip

β(w)w(x˜)Mβ,r,w(f)(x˜),

J≤C

|α|=m 

|Q|

˜

Q

Dαb(x) –Dαb _Q_˜w(x)–/rf(x)w(x)/rdx

≤C |α|=m



|Q|

˜

Q

Dαb(x) –Dαb _Q_˜ rw(x)–rdx /r

˜

Q

(12)

≤C |α|=m



|Q|D α

b_Lip

β(w)w(Q˜)

β/n+/r_w₍_Q˜₎/r–β/n



w(Q˜)–rβ/n

˜

Q

f(x)rw(x)dx /r

≤C |α|=m

Dα_b

Lip_β(w)

w(Q˜)

| ˜Q| Mβ,r,w(f)(x˜)

≤C |α|=m

Dα_b

Lip_β(w)w(x˜)Mβ,r,w(f)(x˜).

ForJ, we have

Rm(b˜;x,y) –Rm(b˜;x,y)

≤C |γ|<m

|α|=m

|x–x|m–|γ||x–y||γ|DαbLip_β(w)w(x˜)w

kQ˜ β/n,

thus

Tb˜(f)(x) –T

˜

b₍_f )(x)

≤ ∞

k=

k+_Q_˜_\_k_Q_˜

Rm(b˜;x,y) –Rm(b˜;x,y)|K (x,y)|

|x–y|mf(y)dy

+

∞

k=

k+_Q_˜_\_k_Q_˜

_|K_x(_–x_y,y_|m) –

K(x,y)

|x–y|m

Rm(b˜;x,y)f(y)dy

+C

|α|=m

∞

k=

k+_Q_˜_\_k_Q_˜

_|K_x(_–x,_yy_|m) –

K(x,y)

|x–y|m

(x–y)αDαb˜(y)f(y)dy

+C

|α|=m

∞

k=

k+_Q˜_\_k_Q˜

(_|xx_––_y|y)mα –

(x–y)α

|x–y|m

K(x,y)Dαb˜(y)f(y)dy

≤C |α|=m

Dαb_Lip

β(w)w(x˜)

∞

k=

wk+Q˜ β/n

×

k+_Q˜_\_k_Q˜

d

(k_d₎n+ +

dε

(k_d₎n+ε

f(y)dy

+C

|α|=m

∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

×

k_Q˜

_Dα

b(y) –Dαb _k_Q_˜w(y)–/rf(y)w(y)/rdy

+C

|α|=m

∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

×

k_Q˜

Dα_b

k_Q˜ –

Dα_b

˜

Qf(y)w(y)

/r_w₍_y₎–/r_dy

≤C |α|=m

Dα_b

Lip_β(w)w(x˜)

× ∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

wkQ˜ β/n

k_Q˜

(13)

×



|kQ|˜

k_Q˜w

(y)–/(r–)_dy

(r–)/r 

|kQ|˜

k_Q˜w (y)dy

/r

k_Q˜_w_k_Q˜ –/r

+C

|α|=m

∞

k=

d

(k_d₎n+ +

dε

(k_d₎n+ε

k_Q˜

Dαb(y) –Dαb _k_Q_˜ r

w(y)–rdy /r

×

k_Q˜

f(y)rw(y)dy /r

+C

|α|=m

Dαb_Lip

β(w)w(x˜)

∞

k=

kwkQ˜ β/n

d

(k_d₎n+ +

dε

(k_d₎n+ε

×

k_Q˜

f(y)rw(y)dx

/r _

|kQ|˜

k_Q˜w(y)

–/(r–)_dy

(r–)/r

×



|kQ|˜

k_Q˜

w(y)dy /r

≤C |α|=m

Dαb_Lip

β(w)w(x˜)

∞

k=

k–k+ –kε 

w(k_Q˜₎–rβ/n

k_Q˜

f(y)rw(y)dx /r

+C

|α|=m

Dαb_Lip

β(w)

× ∞

k=

–k+ –kε w( k_Q˜₎

|k_Q˜_|



w(k_Q˜₎–rβ/n

k_Q_˜

f(y)rw(y)dx /r

≤C |α|=m

_Dα b_Lip

This completes the proof of Theorem .

Proof of Theorem  By putting  <r<pin Theorem  and noticing that w–p _∈_A , by Lemma  and , we obtain

Tb(f)_Lp_(w–p₎≤Mη

Tb(f) _Lp_(w–p₎≤CM#η

Tb(f) _Lp_(w–p₎

≤C |α|=m

Dαb_BMO_(w)wMr,w(f)_Lp_(w–p₎

=C

|α|=m

Dαb_BMO_(w)Mr,w(f)_Lp_(w)

≤C |α|=m

Dαb_BMO_(w)fLp_(w).

This completes the proof of Theorem .

Proof of Theorem By choosing  <r<pin Theorem  and noticing thatw–p_∈_A , by Lemmas  and , we get

Tb(f)_Lp,ϕ_(w–p₎≤Mη

Tb(f) _Lp,ϕ_(w–p₎≤CM#η

Tb(f) _Lp,ϕ_(w–p₎

≤C |α|=m

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=C |α|=m

Dαb_BMO_(w)Mr,w(f)_Lp,ϕ_(w)

≤C |α|=m

Dαb_BMO_(w)fLp,ϕ_(w).

This completes the proof of Theorem .

Proof of Theorem By setting  <r<pin Theorem  and noting thatw–q_∈_A

, by Lem-mas  and , we have

Tb(f)_Lq_(w–q₎≤Mη

Tb(f) _Lq_(w–q₎≤CM #

η

Tb(f) _Lq_(w–q₎

≤C |α|=m

Dα_b

Lip_β(w)wMβ,r,w(f)Lq_(w–q₎

=C

|α|=m

Dαb_Lip

β(w)

Mβ,r,w(f)_Lq_(w)

≤C |α|=m

Dαb_Lip

β(w)fL p_(w).

This completes the proof of Theorem .

Proof of Theorem By choosing  <r<pin Theorem , and noticing thatw–q_∈_A , by Lemmas  and , we get

Tb(f)_Lq,ϕ_(w–q₎≤Mη

Tb(f) _Lq,ϕ_(w–q₎≤CM #

η

Tb(f) _Lq,ϕ_(w–q₎

≤C |α|=m

_Dα b_Lip

β(w)

_wM_β_,r,w(f)_Lq,ϕ_(w–q₎

=C

|α|=m

Dα_b

Lip_β(w)Mβ,r,w(f)Lq,ϕ_(w)

≤C |α|=m

Dα_b

Lip_β(w)fLp,ϕ(w).

This completes the proof of 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.

Author details

1_{Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, People’s Republic of China.}2_{School of}

Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, People’s Republic of China.

Received: 9 January 2014 Accepted: 28 February 2014 Published:18 Mar 2014

References

1. Garcia-Cuerva, J, Rubio de Francia, JL: Weighted Norm Inequalities and Related Topics, vol. 16. North-Holland, Amsterdam (1985)

(15)

3. Torchinsky, A: Real Variable Methods in Harmonic Analysis. Pure and Applied Math., vol. 123. Academic Press, New York (1986)

4. Coifman, RR, Rochberg, R, Weiss, G: Factorization theorems for Hardy spaces in several variables. Ann. Math.103, 611-635 (1976)

5. Pérez, C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal.128, 163-185 (1995) 6. Pérez, C, Trujillo-Gonzalez, R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc.65, 672-692

(2002)

7. Chanillo, S: A note on commutators. Indiana Univ. Math. J.31, 7-16 (1982)

8. Janson, S: Mean oscillation and commutators of singular integral operators. Ark. Math.16, 263-270 (1978) 9. Paluszynski, M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss.

Indiana Univ. Math. J.44, 1-17 (1995)

10. Bloom, S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc.292, 103-122 (1985)

11. Hu, B, Gu, J: Necessary and suﬃcient conditions for boundedness of some commutators with weighted Lipschitz spaces. J. Math. Anal. Appl.340, 598-605 (2008)

12. He, YX, Wang, YS: Commutators of Marcinkiewicz integrals and weighted BMO. Acta Math. Sinica (Chin. Ser.)54, 513-520 (2011)

13. Zhou, XS, Liu, LZ: Necessary and suﬃcient conditions for boundedness of commutators of strongly singular integral operators with weighted Lipschitz functions. Indian J. Pure Appl. Math.42, 405-416 (2011)

14. Cohen, J: A sharp estimate for a multilinear singular integral onRn_{. Indiana Univ. Math. J.}₃₀_{, 693-702 (1981)}

15. Cohen, J, Gosselin, J: On multilinear singular integral operators onRn_{. Stud. Math.}₇₂_{, 199-223 (1982)}

16. Cohen, J, Gosselin, J: A BMO estimate for multilinear singular integral operators. Ill. J. Math.30, 445-465 (1986) 17. Coifman, R, Meyer, Y: Wavelets, Calderón-Zygmund and Multilinear Operators. Cambridge Studies in Advanced

Math., vol. 48. Cambridge University Press, Cambridge (1997)

18. Ding, Y, Lu, SZ: Weighted boundedness for a class rough multilinear operators. Acta Math. Sin.17, 517-526 (2001) 19. Hu, G, Yang, DC: A variant sharp estimate for multilinear singular integral operators. Stud. Math.141, 25-42 (2000) 20. Garcia-Cuerva, J: WeightedHp_{spaces. Diss. Math.}₁₆₂_{, 1-63 (1979)}

21. Peetre, J: On convolution operators leavingLp,λ_{-spaces invariant. Ann. Mat. Pura Appl.}₇₂_{, 295-304 (1966)} 22. Peetre, J: On the theory ofLp,λ_{-spaces. J. Funct. Anal.}₄_{, 71-87 (1969)}

23. Di FaZio, G, Ragusa, MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital.5-A, 323-332 (1991)

24. Di Fazio, G, Ragusa, MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coeﬃcients. J. Funct. Anal.112, 241-256 (1993)

25. Liu, LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math. Sci. Ser. B25, 89-94 (2005)

26. Liu, LZ: Sharp maximal function estimates and boundedness for commutators associated with general integral operator. Filomat25, 137-151 (2011)

27. Liu, LZ: Multilinear singular integral operators on Triebel-Lizorkin and Lebesgue spaces. Bull. Malays. Math. Sci. Soc.

35, 1075-1086 (2012)

28. Mizuhara, T: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis, Proceedings of a Conference Held in Sendai, Japan, pp. 183-189 (1990)

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