Weighted boundedness of a multilinear operator associated to a singular integral operator with general kernels

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R E S E A R C H

Open Access

Weighted boundedness of a multilinear

operator associated to a singular integral

operator with general kernels

Qiufen Feng

*

*_{Correspondence:}

fengqiufen@126.com

Changsha Commerce and Tourism College, Changsha, 410116, P.R. China

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

MSC: 42B20; 42B25

Keywords: multilinear operator; singular integral operator; sharp maximal function; weightedBMO; weighted Lipschitz function

1 Introduction and preliminaries

As the development of singular integral operators (see [–]), their commutators and mul-tilinear operators have been well studied. In [–], the authors prove that the commutators generated by singular integral operators andBMOfunctions are bounded onLp(Rn) for  <p<∞. Chanillo (see []) proves a similar result when singular integral operators are replaced by fractional integral operators. In [, ], the boundedness for the commutators generated by singular integral operators and Lipschitz functions on Triebel-Lizorkin and Lp_(Rn_{) ( <}_p_<_∞_{) spaces is obtained. In [, ], the boundedness for the commutators}

generated by singular integral operators and weightedBMOand Lipschitz functions on

Lp_(Rn_{) ( <}_p_<_∞_{) spaces is obtained. In [], some singular integral operators with}

gen-eral kernel are introduced, and the boundedness for the operators and their commutators generated byBMOand Lipschitz functions is obtained (see [, , ]). Motivated by these, in this paper, we study the multilinear operator generated by the singular integral operator with general kernel and the weighted Lipschitz andBMOfunctions.

First, let us introduce some notations. Throughout this paper,Qdenotes 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,

where, and in what follows,fQ=|Q|–

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

M#(f)(x)≈sup Qxc∈Cinf

 |Q|

Q

f(y) –cdy.

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Let

M(f)(x) =sup Qx

 |Q|

Q

f(y)dy.

Forη> , letM#

η(f)(x) =M

#₍_|_f_|η₎/η_{(x) and}_M

η(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 functions}_f _{such that}

fLp₍_w₎=

Rn

_f(x)pw(x)dx

/p

<∞.

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

bLipβ(w)=sup

Q

 w(Q)β/n

 w(Q)

Q

b(y) –bQ p

w(x)–pdy

/p

<∞,

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

bBMO(w)=sup

Q

 w(Q)

Q

b(y) –bQpw(x)–pdy /p

<∞.

Remark () It has been known that (see [, ]), forb∈Lipβ(w),w∈Aandx∈Q,

|bQ–bk_Q| ≤Ckb_Lip

β(w)w(x)w

kQ β/n.

() It has been known that (see [, ]), forb∈BMO(w),w∈Aandx∈Q,

|bQ–bk_Q| ≤Ckb_BMO₍_w₎w(x).

() Letb∈Lipβ(w) orb∈BMO(w) andw∈A. By [], we know that spacesLipβ(w)

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Deﬁnition  Letϕbe a positive, increasing function onR+_{, and there exists a constant} D>  such that

ϕ(t)≤Dϕ(t) fort≥.

Letwbe a non-negative weight function onRnandfbe a locally integrable function onRn. 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 space (see}

[, ]). Ifϕ(d) = , thenLp,ϕ_(Rn_,_{w) =}_Lp_(Rn_,_{w), which is the weighted Lebesgue space (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 study some singular integral operators as follows (see []).

Deﬁnition  LetT:S→Sbe a linear operator such thatTis bounded onL_(Rn_{) and has}

a kernelK, that is, there exists a locally integrable functionK(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

K(x,y)≤C|x–y|–n,

|y–z|<|x–y|

K(x,y) –K(x,z)+K(y,x) –K(z,x) dx≤C,

and there is a sequence of positive constant numbers{Ck}such that for anyk≥,

k_|z_–_y|≤|x_–_y|_<k+_|z_–_y|

_K(x,_{y) –}_K_(x,_z)₊_K(y,_{x) –}_K(z,_x) q

dy

/q

≤Ck

k|z–y| –n/q,

where  <q<  and /q+ /q= . Moreover, letmbe a positive integer andbbe a function onRn_{. Set}

Rm+(b;x,y) =b(x) –

|α|≤m



α!D

α

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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 Deﬁni-tion  (see [–, , ]) and that the commutator [b,T](f) =bT(f) –T(bf) is a particular operator of the multilinear operatorTb _if_m_{= . The multilinear operator}_Tb_{is a}

non-trivial generalization of the commutator. It is well known that commutators and multilin-ear 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 maxi-mal inequalities for the multilinear operatorTb_{. As application, we obtain the weighted}

Lp-norm inequality and Morrey space boundedness for the multilinear operatorTb.

2 Theorems and lemmas

We shall prove the following theorems.

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A,  <η< ,q<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)M˜ r,w(f)(x).˜

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A,  <η< ,q<r<∞,  <β< and Dα_b_∈_Lip

β(w)for allαwith|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_Lip

β(w)w(x)M˜ β,r,w(f)(x).˜

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A,q<u<∞and Dα_b_∈_BMO(w)_{for all}_α_with_|_α_|₌_m._{Then T}b_{is bounded from}

Lu_(Rn_,_w)_{to L}u_(Rn_,_w–u_).

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A, q<u<∞,  <D< n and Dαb∈BMO(w)for all α with |α|=m. Then Tb is bounded from Lu,ϕ_(Rn_,_w)_{to L}u,ϕ_(Rn_,_w–u_).

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A,  <β< ,q<u<n/β, /v= /u–β/n and Dαb∈Lipβ(w)for allαwith|α|=m.

Then Tbis bounded from Lu(Rn,w)to Lv(Rn,w–v).

Theorem  Let T be a singular integral operator as in Deﬁnition,the sequence{kCk} ∈l,

w∈A,  <β< ,  <D< n_,_q_<_u_<_n/_β_{, /v}_{= /u}_–_β_{/n and D}α_b_∈_Lip

β(w)for allαwith |α|=m.Then Tb_{is bounded from L}u,ϕ_(Rn_,_w)_{to L}v,ϕ_(Rn_,_w–v_).

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Lemma (see [, p.]) Let <u<v<∞and for any function f ≥,we deﬁne that,for /r= /u– /v,

fWLv=sup

λ>

λx∈Rn:f(x) >λ/v, Nu,v(f) =sup Q

fχQLu/χ_Q_Lr,

where thesupis taken for all measurable sets Q with <|Q|<∞.Then

fWLv≤N_u_,_v(f)≤v/(v–u) /uf_WLv.

Lemma (see []) Let T be a singular integral operator as in Deﬁnition,the sequence {Ck} ∈l.Then T is bounded on Lp(Rn,w)for w∈Ap with <p<∞,and weak(L,L)

bounded.

Lemma (see [, ]) Let≤η<n, ≤s<u<n/η, /v= /u–η/n and w∈A.Then

Mη,s,w(f)_Lv₍_w₎≤CfLu₍_w₎.

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

for which the left-hand side is ﬁnite,

RnMη(f)(x)

p_w(x)_dx_≤_C

RnM

#

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_<_u_<_n/_η_{, /v}_{= /u}_–_η_{/n and}

w∈A.Then

Mη,s,w(f)_Lv,ϕ₍_w₎≤CfLu,ϕ₍_w).

Lemma (see []) Let b be a function on Rn_{and D}α_A_∈_Lu_(Rn₎_{for all}_α_with_|_α_|₌_{m and}

any u>n.Then

Rm(b;x,y)≤C|x–y|m

|α|=m

 | ˜Q(x,y)|

˜ Q(x,y)

Dαb(z)udz

/u

,

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

3 Proofs of theorems

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

following inequality holds:

 |Q|

Q

Tb(f)(x) –Cηdx

/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=fχ_Rn_{\ ˜}_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)ηdx

/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)ηdx

/η

=I+I+I.

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

 |Q|

Q

w(x)p_dx

/p

≤ C

|Q|

Q

w(x)dx

for all cubeQand some  <p<∞(see []). We takeu=rp/(r+p– ) in Lemma  and have  <u<randp=u(r– )/(r–u). Then by Lemma  and Hölder’s inequality, we get

Rm(b;˜ x,y)

≤C|x–y|m

|α|=m

 | ˜Q(x,y)|

˜ Q(x,y)

Dα_b(z)˜ u_dz

/u

≤C|x–y|m

|α|=m

| ˜Q|–/u

˜ Q(x,y)

Dαb(z)˜ uw(z)u(–r)/rw(z)u(r–)/rdz

/u

≤C|x–y|m

|α|=m

| ˜Q|–/u

˜ Q(x,y)

Dαb(z)˜ rw(z)–rdz

/r

×

˜ Q(x,y)

w(z)u(r–)/(r–u)dz

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≤C|x–y|m

|α|=m

| ˜Q|–/uDαb_BMO₍_w₎w(Q)˜ /r| ˜Q|(r–u)/ru

×

 | ˜Q(x,y)|

˜ Q(x,y)

w(z)p_dz

(r–u)/ru

≤C|x–y|m

|α|=m

Dαb_BMO₍_w₎| ˜Q|–/uw(Q)˜ /r| ˜Q|/u–/r

 | ˜Q(x,y)|

˜ Q(x,y)

w(z)dz

(r–)/r

≤C|x–y|m

|α|=m _Dα

b_BMO₍_w₎| ˜Q|–/uw(Q)˜ /r| ˜Q|/u–/rw(Q)˜ –/r| ˜Q|/r–

≤C|x–y|m

|α|=m

Dαb_BMO₍_w₎w(Q)˜ | ˜Q|

≤C|x–y|m

|α|=m Dα_b

BMO(w)w(x).˜

Thus, by theLs-boundedness ofT(see Lemma ) for  <s<randw∈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)sdx

/s

≤C

|α|=m _Dα

b_BMO₍_w₎w(x)˜|Q|–/s

Rn

_f(x)s

dx

/s

≤C

|α|=m Dα_b

BMO(w)w(x)˜|Q| –/s

˜ Q

f(x)sw(x)s/rw(x)–s/rdx

/s

≤C

|α|=m Dα_b

BMO(w)w(x)˜|Q| –/s

˜ 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

(r–s)/rs  | ˜Q|

˜ Q

w(x)dx

/r

| ˜Q|/sw(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)ηdx

/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|D

α_b

BMO(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)M˜ r,w(f)(x).˜

ForI, noting 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,_x_–_yy)_|m –

K(x,y) |x–y|m

Rm(b;˜ x,y)f(y)dy

+

|α|=m



α!

Rn

K(x,_|_xy)(x_–_y_|–my)α–

K(x,y)(x–y)α |x–y|m

Dαb(y)˜ f(y)dy

=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)˜

and

Rm(b;˜ x,y)≤C

|α|=m

|x–x|mDαb_BMO₍_w₎w(x).˜

Thus, byw∈A⊆Ar, we get

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|

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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

Dαb_BMO₍_w₎w(x)M˜ r,w(f)(x)˜ ∞

k= –k

≤C

|α|=m

ForI_()(x), we take  <p<∞such that /p+ /q+ /r= . Recallingr>qandw∈A⊆ Ar/p+, we get

I_()(x)≤

∞

k=

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)|

Rm(b;˜ x,y)|

|x–y|m f(y)w(y)

/r_w(y)–/r_dy

+

k+_Q\˜ _k_Q˜

_|_x_–_y_|m –

 |x–y|m

K(x,y)Rm(b;˜ x,y)f(y)dy

≤C

|α|=m Dα_b

BMO(w)w(x)˜

∞

k=

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)

q

dy

/q

×

k+_Q\˜ _k_Q˜

_f(y)rw(y)dy

/r

k+_Q\˜ _k_Q˜

w(y)–p/r_dy /p

+C

|α|=m _Dα

b_BMO₍_w₎w(x)˜

∞

k=

k+_Q\˜ _k_Q˜

|x–x|

|x–y|n+f(y)w(y)

/r_w(y)–/r_dy

≤C

|α|=m Dα_b

BMO(w)w(x)˜

∞

k=

CkkQ

–/q

wk_Q˜ /r

×

 w(k_Q)˜

k_Q_˜

f(y)rw(y)dy

/r  |k_Q˜_|

k_Q_˜w(y)

–p/r_dy /p

×

 |k_Q˜|

k_Q˜w(y)dy

/r

kQ˜/r+/pwkQ˜ –/r

+C

|α|=m

∞

k= d (k_d)n+

×

k_Q˜

f(y)rw(y)dy

/r

k_Q˜

w(y)–/(r–)dy

(10)

≤C

|α|=m

∞

k=

Ck+ –k

 w(k_Q)˜

k_Q˜

f(y)rw(y)dy

/r

≤C

|α|=m

Similarly, we have, forr<p_<∞,  <s,s<∞with /p+ /q+ /r+ /s=  and /q+ /r+ /s= ,

I_()(x)≤

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)|(x–y)

α_|

|x–y|m

×Dαb(y) –Dαb _k+_Q_˜w(y)(–p)/pf(y)w(y)/rw(y)(p–)/p–/rdy

+

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

_K_(x,_{y) –}_K(x,_y)|(x–y)α_| |x–y|m

×Dαb _k+_Q˜ –

Dαb _Q_˜f(y)w(y)/rw(y)–/rdy

+

|α|=m

k+_Q\_˜ _k_Q_˜

(x_|_x_––_yy)_|mα –

(x–y)α |x–y|m

K(x,y)f(y)Dα_b(y)˜ _dy

≤C

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)qdy

/q

k+_Q˜

f(y)rw(y)dy

/r

×

k+_Q˜

Dα_{b(y) –}_Dα_b

k+_Q˜

p

w(y)–p_dy

/p

×

k+_Q_˜w(y)

–(/r–/p_)s_dy

/s

+C

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)

q

dy

/q

×

k+_Q_˜

f(y)rw(y)dy

/r

×Dαb _k+_Q_˜ –

Dαb _Q_˜

k+_Q˜w(y) –s/r_dy

/s

+C

|α|=m ∞

k=

k+_Q\_˜ _k_Q_˜

|x–x| |x–y|n+

×f(y)Dα_{b(y) –}_Dα_b

k+_Q˜w(y)/rw(y)–/rdy

+C

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

|x–x| |x–y|n+

×f(y)Dαb _k+_Q_˜ –

Dαb _Q_˜w(y)/rw(y)–/rdy

≤C

|α|=m ∞

k=

CkkQ

–/q

Dαb_BMO₍_w₎wkQ˜ /pwkQ˜ /r

×

 w(k_Q)˜

k_Q˜

f(y)rw(y)dy

(11)

×

 |k_Q˜|

k_Q˜w(y)dy

/r–/p_

 |k_Q˜|

k_Q˜ w(y)

–(/r–/p)s_dy

/s

×kQ˜/s+/r–/p_w_k_Q˜ –/r+/p

+C

|α|=m ∞

k=

CkkQ

–/q

kDαb_BMO₍_w₎w(x)w˜ kQ˜ /r

×

 w(k_Q)˜

k_Q˜

f(y)rw(y)dy

/r  |k_Q˜_|

k_Q˜

w(y)dy

/r

×

 |k_Q˜_|

k_Q_˜w(y)

–s/r_dy

/s

kQ˜/s+/rwkQ˜ –/r

+C

|α|=m ∞

k= d (k_d)n+

k_Q_˜

Dαb(y) –Dαb _k_Q˜

r

w(y)–rdy

/r

×

k+_Q˜

f(y)rw(y)dy

/r

+C

|α|=m ∞

k= d (k_d)n+kD

α

b_BMO₍_w₎w(x)˜

k_Q_˜w(y)

–r/r_dy /r

×

k+_Q˜

f(y)rw(y)dy

/r

≤

|α|=m

Dαb_BMO₍_w₎

∞

k= Ck

w(kQ)˜ |k_Q˜_|

 w(k_Q)˜

k_Q_˜

f(y)rw(y)dy

/r

+

|α|=m Dα_b

BMO(w)w(x)˜

∞ k= kCk  w(k_Q)˜

k_Q˜

f(y)rw(y)dy

/r

+C

|α|=m

Dαb_BMO₍_w₎

∞

k= –kw(

k_Q)˜

|k_Q˜|

 w(k_Q)˜

k_Q˜

f(y)rw(y)dx

/r

+C

|α|=m

∞

k= k–k

 w(k_Q)˜

k_Q_˜

f(y)rw(y)dx

/r

≤C

|α|=m

Thus

I≤C

|α|=m Dα_b

BMO(w)w(x)M˜ r,w(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|

Q

Tb(f)(x) –Cηdx

/η

≤C

|α|=m

Dαb_Lip

(12)

Fix a cubeQ=Q(x,d) andx˜∈Q. Similar to the proof of Theorem , we have, forf=fχQ˜

andf=fχ_Rn_{\ ˜}_Q,

 |Q|

Q

Tb(f)(x) –Tb˜(f)(x)ηdx

/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)ηdx

/η

=J+J+J.

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)

_Dα˜

b(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

Lipβ(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

Dαb_Lip β(w)| ˜Q|

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

≤C|x–y|m

|α|=m

Dαb_Lip β(w)

w(Q)˜ β/n+ | ˜Q|

≤C|x–y|m |α|=m

Dα_b

Lipβ(w)w(Q)˜ β/n_w(_x)._˜

Thus

J≤C

|α|=m

Dαb_Lip

β(w)w(Q)˜

β/n_w(_x)_˜ _|_Q_|–/s

Rn

f(x)sdx

/s

≤C

|α|=m

Dαb_Lip

β(w)w(Q)˜

(13)

×

˜ 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

(r–s)/rs  | ˜Q|

˜ Q

w(x)dx

/r

| ˜Q|/sw(Q)˜ –/r

≤C

|α|=m

Dαb_Lip

β(w)w(x)M˜ β,r,w(f)(x),˜

J≤C

|α|=m

 |Q|

˜ Q

_Dα_{b(x) –}

Dαb _Q_˜w(x)–/r_f_(x)_w(x)/r_dx

≤C

|α|=m

 |Q|

˜ Q

Dαb(x) –Dαb _Q_˜ rw(x)–rdx

/r

˜ Q

f(x)rw(x)dx

/r

≤C

|α|=m

 |Q|D

α_b

Lipβ(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

Dαb_Lip β(w)

w(Q)˜

| ˜Q| Mβ,r,w(f)(x)˜

≤C

|α|=m

Dαb_Lip

β(w)w(x)M˜ β,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|Dαb_Lip

β(w)w(x)w˜

kQ˜

and

Rm(b;˜ x,y)≤C

|α|=m

|x–x|mDαb_Lip

β(w)w(x)w˜

kQ˜ .

Thus, for  <p<∞with /p+ /q+ /r=  andr<p_<∞,  <s,s<∞with /p+ /q+ /r+ /s=  and /q+ /r+ /s= , we obtain

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,_x_–_yy)_|m –

K(x,y) |x–y|m

Rm(b;˜ x,y)f(y)dy

+

|α|=m



α!

Rn

K(x,_|_xy)(x_–_y_|–my)α–

K(x,y)(x–y)α |x–y|m

Dαb(y)˜ f(y)dy

≤C

|α|=m

Dαb_Lip β(w)w(x)˜

∞

k=

wk+Q˜ β/n

k+_Q\_˜ _k_Q_˜

|x–x|

(14)

+C

|α|=m

∞

k=

wk+Q˜ β/n

k+_Q\˜ _k_Q˜

K(x,y) –K(x,y)qdy

/q

×

k+_Q\˜ _k_Q˜

_f(y)rw(y)dy

/r

k+_Q\˜ _k_Q˜

w(y)–p/r_dy /p

+C

|α|=m ∞

k=

k+_Q\_˜ _k_Q_˜

|x–x| |x–y|n+

Dαb(y) –Dαb _k+_Q˜f(y)dy

+C

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

|x–x| |x–y|n+D

α

b _k+_Q_˜ –

Dαb _Q_˜f(y)dy

+C

|α|=m ∞

k=

k+_Q\˜ _k_Q˜

/q

k+_Q˜

f(y)rw(y)dy

/r

×

k+_Q˜

Dαb(y) –Dαb _k+_Q_˜

p

w(y)–p_dy

/p

k+_Q˜w(y)

–(/r–/p)s_dy

/s

+C

|α|=m ∞

k=

k+_Q\_˜ _k_Q_˜

/q

k+_Q_˜

f(y)rw(y)dy

/r

×Dαb _k+_Q_˜ –

Dαb _Q_˜

k+_Q˜w(y) –s/r_dy

/s

≤C

|α|=m

∞

k=

k d

(k_d)n+w

kQ˜ β/n

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

Lipβ(w)w(x)˜

∞

k=

CkkQ

–/q

wkQ˜ β/n

k_Q˜

f(y)rw(y)dx

/r

×

 |k_Q˜_|

k_Q_˜w(y)

–p/r_dy /p

 |k_Q˜_|

k_Q_˜w(y)dy

/r

kQ˜/p+/rwkQ˜ –/r

+C

|α|=m ∞

k= d (k_d)n+

k_Q˜

Dα_{b(y) –}_Dα_b

k_Q˜

r

w(y)–rdy

/r

×

k_Q˜

f(y)rw(y)dy

/r

+C

|α|=m

∞

k=

CkkQ

–/q

wkQ˜ β/n

k_Q˜

f(y)rw(y)dx

/r

×

 |k_Q˜_|

k_Q_˜w(y)

–(/r–/p_)s_dy

/s _

|k_Q˜_|

k_Q_˜ w(y)dy

/r–/p_

×k_Q˜/s+/r–/p_w_k_Q˜ –/r+/p

+C

|α|=m Dα_b

Lipβ(w)w(x)˜

∞

k=

kCkkQ

–/q

wkQ˜ β/n

k_Q_˜

f(y)rw(y)dy

/r

×

 |k_Q˜_|

k_Q˜w(y)dy

/r  |k_Q˜_|

k_Q˜ w(y)

–s/r_dy

/s

(15)

≤C

|α|=m

∞

k=

kCk+ –k

×

 w(k_Q)˜ –rβ/n

k_Q˜

_f(y)rw(y)dx

/r

+C

|α|=m

Dαb_Lip β(w)

∞

k=

kCk+ –k

w(kQ)˜ |k_Q˜|

×

 w(k_Q)˜ –rβ/n

k_Q˜

_f_(y)r

w(y)dx

/r

≤C

|α|=m Dα_b

Lipβ(w)w(x)M˜ β,r,w(f)(x).˜

This completes the proof of Theorem .

Proof of Theorem Choose  <r<uin Theorem  and noticew–u_∈_{A, then we have, by}

Lemmas  and ,

Tb(f)_Lu₍_w–u₎≤Mη

Tb(f) _Lu₍_w–u₎≤CM

#

η

Tb(f) _Lu₍_w–u₎

≤C

|α|=m Dα_b

BMO(w)wMr,w(f)Lu₍_w–u₎

=C

|α|=m

Dαb_BMO₍_w₎Mr,w(f)_Lu₍_w₎

≤C

|α|=m

Dαb_BMO₍_w₎fLu₍_w).

This completes the proof of Theorem .

Proof of Theorem Choose  <r<uin Theorem  and noticew–u_∈_{A, then we have, by}

Lemmas  and ,

Tb(f)_Lu,ϕ₍_w–u₎≤Mη

Tb(f) _Lu,ϕ₍_w–u₎≤CM#η

Tb(f) _Lu,ϕ₍_w–u₎

≤C

|α|=m

Dαb_BMO₍_w₎wMr,w(f)_Lu,ϕ₍_w–u₎

=C

|α|=m

Dαb_BMO₍_w₎Mr,w(f)_Lu,ϕ₍_w₎

≤C

|α|=m

Dαb_BMO₍_w₎fLu,ϕ₍_w).

This completes the proof of Theorem .

Proof of Theorem Choose  <r<uin Theorem  and noticew–v∈A, then we have, by Lemmas  and ,

Tb(f)_Lv₍_w–v₎≤Mη

Tb(f) _Lv₍_w–v₎≤CM

#

η

Tb(f) _Lv₍_w–v₎

≤C

|α|=m

Dαb_Lip β(w)

(16)

=C

|α|=m

Dαb_Lip β(w)

Mβ,r,w(f)_Lv₍_w₎

≤C

|α|=m

Dαb_Lip β(w)fL

u₍_w).

This completes the proof of Theorem .

Proof of Theorem Choose  <r<uin Theorem  and noticew–v_∈_{A, then we have, by}

Lemmas  and ,

Tb(f)_Lv,ϕ₍_w–v₎≤Mη

Tb(f) _Lv,ϕ₍_w–v₎≤CM

#

η

Tb(f) _Lv,ϕ₍_w–v₎

≤C

|α|=m _Dα

b_Lip β(w)

_wM_β_,_r_,_w(f)_Lv,ϕ₍_w–v₎

=C

|α|=m Dα_b

Lipβ(w)

Mβ,r,w(f)_Lv,ϕ₍_w₎

≤C

|α|=m Dα_b

Lipβ(w)fL

u,ϕ₍_w₎.

This completes the proof of Theorem .

Competing interests

The author declares that they have no competing interests.

Received: 24 December 2013 Accepted: 25 April 2014 Published:13 May 2014

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