Weighted boundedness of multilinear operators associated to singular integral operators with non smooth kernels

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R E S E A R C H

Open Access

Weighted boundedness of multilinear

operators associated to singular integral

operators with non-smooth kernels

Daqing Lu

*

*_{Correspondence:}

ldqcslgdx@163.com

College of Mathematics, Changsha University of Science and Technology, Changsha, 410077, 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 non-smooth kernel. 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 the singular integral operators andBMOfunctions are bounded onLp₍_Rn₎

for  <p<∞. Chanillo (see []) proves a similar result when the singular integral op-erators are replaced by the fractional integral opop-erators. In [, ], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin andLp₍_Rn_{) ( <}_p_<_∞_{) spaces is obtained. In [, ], the boundedness for}

the commutators generated by the singular integral operators and the weightedBMOand Lipschitz functions onLp(Rn) ( <p<∞) spaces is obtained (also see []). In [, ], some singular integral operators with non-smooth kernels are introduced, and the boundedness for the operators and their commutators is obtained (see [–]). Motivated by these, in this paper, we study multilinear operators generated by singular integral operators with non-smooth kernels and the weighted Lipschitz andBMOfunctions.

In this paper, we study some singular integral operators as follows (see []).

Deﬁnition  A family of operatorsDt,t> , is said to be an ‘approximation to the identity’

if, for everyt> ,Dtcan be represented by a kernelat(x,y) in the following sense:

Dt(f)(x) =

Rnat(x,y)f(y)dy

for everyf ∈Lp₍_Rn_{) with}_p_≥_{, and}_a

t(x,y) satisﬁes

at(x,y)≤ht(x,y) =Ct–n/ρ

|x–y|/t,

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whereρis a positive, bounded and decreasing function satisfying

lim r→∞r

n+

ρr= 

for some> .

Deﬁnition  A linear operatorT is called a singular integral operator with non-smooth kernel ifTis bounded onL₍_Rn_{) and associated with the kernel}_K₍_x_,_y_{) so that}

T(f)(x) =

RnK(x,y)f(y)dy

for every continuous function f with compact support, and for almost allx not in the support off.

() There exists an ‘approximation to the identity’{Bt,t> }such thatTBthas the

asso-ciated kernelkt(x,y) and there existc,c>  so that

|x–y|>ct/

K(x,y) –kt(x,y)dx≤c for ally∈Rn.

() There exists an ‘approximation to the identity’{At,t> }such thatAtT has the

as-sociated kernelKt(x,y) which satisﬁes

Kt(x,y)≤ct–n/ if|x–y| ≤ct/

and

K(x,y) –Kt(x,y)≤ctδ/|x–y|–n–δ if|x–y| ≥ct/

for someδ> ,c,c> . Moreover, letmbe a positive integer andbbe a 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 commutator [b,T](f) =bT(f) –T(bf) is a particular operator of the multi-linear operatorTb_if_m_{= . The multilinear operator}_Tb_{is a non-trivial generalization 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 sharp maximal inequalities for the multilinear operatorTb_{. As an application, we obtain the weighted}_Lp_{-norm inequality and Morrey}

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Now, let us introduce some notations. Throughout this paper,Qwill denote a cube of

Rn_{with sides parallel to the axes. For any locally integrable function}_f_{, 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.

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= .

The sharp maximal functionMA(f) associated with the ‘approximation to the identity’ {At,t> }is deﬁned by

M#_A(f)(x) =sup x∈Q



|Q|

Q

f(y) –AtQ(f)(y)dy,

wheretQ=l(Q)andl(Q) denotes the side length ofQ. Forη> , letMA#,η(f) =M

#

A(|f|

η₎/η_.

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}

f Lp₍_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

b Lipβ(w)=sup

Q



w(Q)β/n



w(Q)

Q

b(y) –bQpw(x)–pdy

/p

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

b BMO(w)=sup

Q



w(Q)

Q

b(y) –bQ p

w(x)–pdy

/p

<∞.

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

|bQ–bk_Q| ≤Ck b _Lip_β₍_w₎w(x)w

kQβ/n.

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

|bQ–bk_Q| ≤Ck b _BMO₍_w₎w(x).

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

orBMO(w) coincide and the norms b Lipβ(w)or b BMO(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 ≤η<nand ≤p<n/η,

f Lp,η,ϕ₍_w₎= sup

x∈Rn_,_d_>



ϕ(d)–pη/n

Q(x,d)

f(y)pw(y)dy

/p

,

whereQ(x,d) ={y∈Rn:|x–y|<d}. The generalized fractional weighted Morrey space is deﬁned by

Lp,η,ϕRn,w=f ∈L_locRn: f Lp,η,ϕ₍_w₎<∞.

We writeLp,η,ϕ₍_Rn_{) =}_Lp,ϕ₍_Rn_{) if}_η_{= , which is the generalized weighted Morrey space.}

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 [, –]).

2 Theorems and lemmas

We shall prove the following theorems.

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exists a constant C> such that,for any f∈C_∞(Rn₎_and_x_˜_∈_Rn_,

M#_A_,η

Tb(f)(x˜)≤C |α|=m

Dαb_BMO₍_w₎w(x˜)Mr,w(f)(x˜).

Theorem  Let T be a singular integral operator with non-smooth kernel as given in Def-inition,w∈A,  <η< ,  <r<∞,  <β< and Dαb∈Lipβ(w)for allαwith|α|=m.

Then there exists a constant C> such that,for any f ∈C∞_ (Rn₎_and_x_˜_∈_Rn_,

M#_A_,_ηTb(f)(x˜)≤C |α|=m

Dα_b

Lipβ(w)w(x˜)Mβ,r,w(f)(x˜).

Theorem  Let T be a singular integral operator with non-smooth kernel as given in Deﬁ-nition,w∈A,  <p<∞and Dαb∈BMO(w)for allαwith|α|=m.Then Tbis bounded

from Lp₍_Rn_,_w₎_{to L}p₍_Rn_,_w–p_).

Theorem  Let T be a singular integral operator with non-smooth kernel as given in Def-inition,w∈A,  <p<∞,  <D< nand Dαb∈BMO(w)for allαwith|α|=m.Then Tb

is bounded from Lp,ϕ₍_Rn_,_w₎_{to L}p,ϕ₍_Rn_,_w–p_).

Theorem  Let T be a singular integral operator with non-smooth kernel as given in Def-inition,w∈A,  <β< ,  <p<n/β, /q= /p–β/n and Dαb∈Lipβ(w)for allαwith

|α|=m.Then Tb_{is bounded from L}p₍_Rn_,_w₎_{to L}q₍_Rn_,_w–q_).

Theorem  Let T be a singular integral operator with non-smooth kernel as given in Def-inition,w∈A,  <β< ,  <D< n,  <p<n/β, /q= /p–β/n and Dαb∈Lipβ(w)for

allαwith|α|=m.Then Tb_{is bounded from L}p,β,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 that,for

/r= /p– /q,

f WLq=sup

λ>

λx∈Rn:f(x) >λ/q, Np,q(f) =sup Q

fχQ Lp/ χ_Q _Lr,

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

f WLq≤N_p_,_q(f)≤q/(q–p)/p f _WLq.

Lemma (see [, ]) Let T be a singular integral operator with non-smooth kernel as given in Deﬁnition.Then T is bounded on Lp₍_Rn_,_w₎_{for w}_∈_A

pwith <p<∞,and weak

(L,L)bounded.

Lemma ([, ]) Let{At,t> }be an‘approximation to the identity’.For anyγ > , there exists a constant C> independent ofγ such that

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forλ> ,where D is a ﬁxed constant which only depends on n.Thus,for f ∈Lp₍_Rn_{),  <}_p_< ∞,  <η<∞and w∈A,

Mη(f)_Lp₍_w₎≤CM #

A,η(f)_Lp₍_w₎.

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

Mη,s,w(f)_Lq₍_w₎≤C f Lp₍_w₎.

Lemma (see [, ]) Let{At,t> }be an‘approximation to the identity’,  <D< n,

 <p<∞,  <η<∞,w∈Aand w∈A.Then

Mη(f)_Lp,ϕ₍_w₎≤CM #

A,η(f)_Lp,ϕ₍_w₎.

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

Mη,s,w(f)_Lq,ϕ₍_w₎≤C f Lp,η,ϕ₍_w₎.

Lemma (see []) Let b be a function on Rn_{and D}α_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)

Dα_b₍_z₎q_dz

/q

,

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

Lemma  Let{At,t> }be an‘approximation to the identity’,w∈Aand b∈BMO(w).

Then,for every f ∈Lp₍_w_),_p_{> ,  <}_r_<_∞_and_x_˜_∈_Rn_,

sup Q˜x



|Q|

Q

AtQ

(b–bQ)f

(y)dy≤C b BMO(w)w(x˜)Mr,w(f)(x˜),

where tQ=l(Q)and l(Q)denotes the side length of Q.

Proof We write, for any cubeQwithx˜∈Q,



|Q|

Q

AtQ

(b–bQ)f

(x)dx

≤ 

|Q|

Q

RnhtQ(x,y)

b(y) –bQ

f(y)dy dx ≤_|Q|

Q

htQ(x,y)b(y) –bQ

f(y)dy dx

+

∞

k= 

|Q|

Q

k+_Q\_k_QhtQ(x,y)

b(y) –bQ

f(y)dy dx

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We have, by Hölder’s inequality,

I≤ C

|Q||Q|

Q

b(y) –bQ

f(y)dy dx ≤_|Q|C

Q

_b₍_y_{) –}_b_Q_w₍_y₎–/r_f₍_y₎_w₍_y₎/r_dy

≤_|Q|C

Q

b(y) –bQ r

w(y)–rdy

/r Q

f(y)rw(y)dy

/r

≤_|Q|C b BMO(w)w(Q)/r

w(Q)/r



w(Q)

Q

f(y)rw(y)dy

/r

≤C b BMO(w)

w(Q)

|Q| Mr,w(f)(x˜) ≤C b BMO(w)w(x˜)Mr,w(f)(x˜).

ForII, notice forx∈Qandy∈k+Q\kQ, then|x–y| ≥k–tQandhtQ(x,y)≤C

s((k–)₎ |Q| ,

then

II≤C ∞

k=

s(k–) 

|Q||Q|

Q

k+_Q

b(y) –bQ

f(y)dy dx

≤C

∞

k=

kns(k–) 

|k+_Q|

×

k+_Q

b(y) –b_k+_Q

+ (b_k+_Q–bQ)f(y)dy

≤C

∞

k=

kns(k–)k+Q–

k+_Q

b(y) –b_k+_Qr

w(y)–rdy

/r

×

k+_Q

_f₍_y₎r w(y)dy

/r

+C ∞

k=

kns(k–)k+Q–k b BMO(w)w(x˜)

k+_Q

f(y)rw(y)dy

/r

×



|k+Q|

k+_Qw

(y)–/(r–)_dy (r–)/r

×



|k+_Q|

k+_Qw(y)dy

/r

k+Qwk+Q–/r

≤C b BMO(w)

∞

k=

kkn_s_(k–)w(k+Q)

|k+Q| +w(x˜)

×



w(k+_Q₎

k+_Q

f(y)rw(y)dy

/r

≤C b BMO(w)

∞

k=

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where the last inequality follows from

∞

k=

k(k–)ns(k–)≤C ∞

k=

k–(k–)_<_∞

for some> . This completes the proof.

Lemma  Let{At,t> }be an‘approximation to the identity’,w∈A,  <β< ,  <r<∞

and b∈Lipβ(w).Then,for every f ∈Lp(w),p> andx˜∈Rn,

sup Q˜x



|Q|

Q

AtQ

(b–bQ)f

(y)dy≤C b Lipβ(w)w(x˜)Mβ,w,r(f)(x˜).

The same argument as in the proof of Lemma  will give the proof of Lemma , we omit the details.

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) –AtQ

Tb(f)(x)ηdx

/η

≤C

|α|=m

Dαb_BMO₍_w₎w(x˜)Mr,w(f)(x˜),

wheretQ=d andddenotes the side length ofQ. Fix a cubeQ=Q(x,d) andx˜∈Q. Let

˜

Q= √nQandb˜(x) =b(x) –_|_α_|₌_m 

α!(D

α_b₎

˜

Qxα, thenRm(b;x,y) =Rm(b˜;x,y) andDαb˜= Dα_b_{– (}_Dα_b₎

˜

Qfor|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 +T ˜ b₍_f

)(x)

and

AtQT

b₍_f₎₍_x_{) =}

Rn

Rm(b˜j;x,y)

|x–y|m Kt(x,y)f(y)dy

–

|α|=m



α!

Rn

(x–y)α_Dα_b˜₍_y₎

|x–y|m Kt(x,y)f(y)dy

+

Rn

Rm+(b˜;x,y)

|x–y|m Kt(x,y)f(y)dy

=AtQT

Rm(b˜;x,·)

|x–·|m f –AtQT

|α|=m



α!

(x–·)α_Dα_b˜

|x–·|m f +AtQT

˜ b₍_f

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then



|Q|

Q

Tb(f)(x) –AtQT

b₍_f₎₍_x₎η

dx /η ≤C  |Q| Q T

Rm(b˜;x,·) |x–·|m f (x)

ηdx /η +C  |Q| Q T

|α|=m



α!

(x–·)α_Dα_b˜

|x–·|m f (x)

ηdx /η +C  |Q| Q AtQT

Rm(b˜;x,·) |x–·|m f (x)

ηdx /η +C 

|Q|

Q

AtQT

|α|=m



α!

(x–·)α_Dα_b˜

|x–·|m f (x)

ηdx /η +C  |Q| Q

T˜b(f)(x) –AtQT

˜ b₍_f

)(x)

η

dx

/η

=I+I+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 Lemma  and Hölder’s inequality, we get

Rm(b˜;x,y)≤C|x–y|m

|α|=m



| ˜Q(x,y)|

˜ Q(x,y)

Dαb˜(z)qdz

/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)

Dα_b˜₍_z₎r_w₍_z₎–r_dz

/r

×

˜ Q(x,y)

w(z)q(r–)/(r–q)dz

(r–q)/rq

≤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

_Dα

b_BMO₍_w₎| ˜Q|–/qw(Q˜)/r_{| ˜}_Q|/q–/r

×



| ˜Q(x,y)|

˜ Q(x,y)

w(z)dz

(10)

≤C|x–y|m

|α|=m

Dαb_BMO₍_w₎| ˜Q|–/qw(Q˜)/r| ˜Q|/q–/rw(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 ofT(see Lemma ) for  <s<randw∈A⊆Ar/s, we obtain

I≤

C |Q| Q T

Rm(b˜;x,·) |x–·|m f (x)

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

_Dα

b_BMO₍_w₎w(x˜)|Q|–/s

˜ Q

_f₍_x₎s

w(x)s/rw(x)–s/rdx

/s

≤C

|α|=m

Dαb_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

Dα_b

BMO(w)w(x˜)|Q|

–/s_w₍_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|/sw(Q˜)–/r

≤C

|α|=m

Dα_b

BMO(w)w(x˜)Mr,w(f)(x˜).

ForI, by the weak (L,L) boundedness ofT(see Lemma ) and Kolmogorov’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˜

)χQ Lη

χQ Lη/(–η)

≤C

|α|=m



|Q|T

Dαbf˜ _WL

≤C

|α|=m



|Q|

Rn

_Dα˜

b(x)f(x)dx

≤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

f(x)rw(x)dx

(11)

≤C |α|=m



|Q|D

α

b_BMO₍_w₎w(Q˜)/rw(Q˜)/r



w(Q˜)

˜ Q

f(x)rw(x)dx

/r

≤C

|α|=m

Dα_b

BMO(w)

w(Q˜)

| ˜Q| Mr,w(f)(x˜)

≤C

|α|=m

Dα_b

ForIandI, by Lemma  and similar to the proof ofIandI, we get

I≤

C |Q|

Q

T

Rm(b˜;x,·) |x–·|m f (x)

dx

≤C

|α|=m

Dα_b

BMO(w)w(x˜)Mr,w(f)(x˜),

I≤C

|α|=m



|Q|

Q

TDαbf˜

(x)ηdx

/η

≤C

|α|=m

Dαb_BMO₍_w₎w(x˜)Mr,w(f)(x˜).

ForI, note that|x–y| ≈ |x–y|forx∈Qandy∈Rn\Q. We have, by Lemma  and similar to the proof ofI,

_R_m₍b˜;x,y)≤C|x–y|m

|α|=m

_Dα

b_BMO₍_w₎w(x˜).

Thus, by the conditions onKandKt, andw∈A⊆Ar,

Tb˜(f)(x) –AtQT

˜ b₍_f

)(x)

≤

Rn

|Rm(b˜;x,y)|

|x–y|m K(x,y) –Kt(x,y)f(y)dy

+

|α|=m



α!

Rn

|Dα_b˜

(y)||(x–y)α|

|x–y|m K(x,y) –Kt(x,y)f(y)dy

≤ ∞

k=

|α|=m

k+_Q\_˜ _k_Q_˜

dδ

|x–y|n+δ

f(y)w(y)/rw(y)–/rdy

+C |α|=m

∞

k=

k+_Q˜

Dα_b

k+_Q˜ –

Dα_b

˜ Q

dδ

|x–y|n+δ

+C |α|=m

∞

k=

k+_Q˜

Dαb(y) –Dαb_k+_Q˜ dδ

|x–y|n+δ

≤C

|α|=m

∞

k=

k d

δ

(k_d₎n+δ

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

(12)

+C |α|=m

∞

k=

dδ

(k_d₎n+δ

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–kδ



w(k_Q˜₎

k_Q˜

f(y)rw(y)dx

/r

+C |α|=m

Dαb_BMO₍_w₎ ∞

k=

–kδw(

k_Q˜₎

|k_Q˜_|



w(k_Q˜₎

k_Q_˜

f(y)rw(y)dx

/r

≤C

|α|=m

Dα_b

BMO(w)w(x˜)Mr,w(f)(x˜)

∞

k=

k–kδ

≤C

|α|=m

Dα_b

Thus

I≤C

|α|=m

_Dα

b_BMO₍_w₎w(x˜)Mr,w(f)(x˜).

These complete the proof of Theorem .

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

that the following inequality holds:



|Q|

Q

Tb(f)(x) –AtQ

Tb(f)(x)ηdx

/η

≤C

|α|=m

Dα_b

Lipβ(w)w(x˜)Mβ,r,w(f)(x˜),

wheretQ=dandddenotes the side length ofQ. 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) –AtQT

b₍_f₎₍_x₎η

dx /η ≤  |Q| Q T

Rm(b˜;x,·) |x–·|m f (x)

ηdx /η +  |Q| Q T

|α|=m



α!

(x–·)α_Dα_b˜

|x–·|m f (x)

ηdx /η + 

|Q|

Q

AtQT

Rm(b˜;x,·) |x–·|m f (x)

ηdx /η + 

|Q|

Q

AtQT

|α|=m



α!

(x–·)α_Dα_b˜

|x–·|m f (x)

ηdx /η +  |Q| Q

Tb˜₍_f

)(x) –AtQT

˜ b₍_f

)(x)

η

dx

/η

(13)

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)

Dα_b˜₍_z₎q_w₍_z₎q(–r)/r_w₍_z₎q(r–)/r_dz

/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

Lipβ(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

Dαb_Lip

β(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)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

_Dα

b_Lip

β(w)w(x˜)| ˜Q|

–/s_w₍_Q˜₎/r



w(Q˜)–rβ/n

˜ Q

_f₍_x₎r w(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

f(x)rw(x)dx

(14)

≤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

Lipβ(w)

w(Q˜)

| ˜Q| Mβ,r,w(f)(x˜)

≤C

|α|=m

Dα_b

ForJandJ, by Lemma  and similar to the proof ofJandJ, we get

J≤

C |Q|

Q

T

Rm(b˜;x,·) |x–·|m f (x)

dx

≤C

|α|=m

Dαb_Lip

β(w)w(x˜)Mβ,r,w(f)(x˜),

J≤C

|α|=m



|Q|

Q

TDα_bf˜



(x)ηdx

/η

≤C

|α|=m

Dα_b

ForJ, by Lemma  and similar to the proof ofJ, fork≥, we have

Rm(b˜;x,y)≤C|x–y|m

|α|=m

Dαb_Lip

β(w)w

kQ˜β/nw(x˜).

Thus

Tb˜(f)(x) –AtQT

˜ b₍_f

)(x)

≤

Rn

|Rm(b˜;x,y)|

|x–y|m K(x,y) –Kt(x,y)f(y)dy

+

|α|=m



α!

Rn

|Dα_b˜

(y)||(x–y)α|

|x–y|m K(x,y) –Kt(x,y)f(y)dy

≤ ∞

k=

|α|=m

Dαb_Lip

β(w)w(x˜)w

kQ˜β/n

×

k+_Q\˜ _k_Q˜

dδ

|x–y|n+δ

+C |α|=m

∞

k=

k+_Q˜

Dα_b

k+_Q˜ –

Dα_b

˜ Q

dδ

|x–y|n+δ

+C |α|=m

∞

k=

k+_Q˜

Dαb(y) –Dαb_k+_Q˜ dδ

|x–y|n+δ

≤C

|α|=m

Dαb_Lip

β(w)w(x˜)

∞

k=

k d

δ

(k_d₎n+δw

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

(15)

+C |α|=m

∞

k=

dδ

(k_d₎n+δ

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=

k–kδ



w(k_Q˜₎–rβ/n

k_Q˜

f(y)rw(y)dx

/r

+C |α|=m

Dαb_Lip

β(w)

∞

k=

–kδw(

k_Q˜₎

|k_Q˜_|



w(k_Q˜₎–rβ/n

k_Q_˜

f(y)rw(y)dx

/r

≤C

|α|=m

Dαb_Lip

β(w)w(x˜)Mβ,r,w(f)(x˜).

This completes the proof of Theorem .

Proof of Theorem Choose  <r<pin Theorem  and noticew–p∈A, then we have, by Lemmas  and ,

Tb₍_f₎

Lp₍_w–p₎≤Mη

Tb₍_f₎

Lp₍_w–p₎≤CM#_A_,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₎ f Lp₍_w₎.

This completes the proof of Theorem .

Proof of Theorem Choose  <r<pin Theorem  and noticew–p_∈_A

, then we have, by Lemmas  and ,

Tb(f)_Lp,ϕ₍_w–p₎≤Mη

Tb(f)_Lp,ϕ₍_w–p₎≤CM #

A,η

Tb(f)_Lp,ϕ₍_w–p₎

≤C

|α|=m

_Dα

b_BMO₍_w₎wMr,w(f)_Lp,ϕ₍_w–p₎

=C |α|=m

Dα_b

BMO(w)Mr,w(f)Lp,ϕ₍_w₎

≤C

|α|=m

Dα_b

BMO(w) f Lp,ϕ(w).

This completes the proof of Theorem .

Proof of Theorem Choose  <r<pin Theorem  and noticew–q_∈_A

, then we have, by Lemmas  and ,

Tb(f)_Lq₍_w–q₎≤Mη

Tb(f)_Lq₍_w–q₎≤CM#_A_,η

Tb(f)_Lq₍_w–q₎

≤C

|α|=m

Dαb_Lip

β(w)

(16)

=C |α|=m

Dαb_Lip

β(w)

Mβ,r,w(f)_Lq₍_w₎

≤C

|α|=m

Dαb_Lip

β(w) f L p₍_w₎.

This completes the proof of Theorem .

Proof of Theorem Choose  <r<pin Theorem  and noticew–q_∈_A

, then we have, by Lemmas  and ,

Tb(f)_Lq,ϕ₍_w–q₎≤Mη

Tb(f)_Lq,ϕ₍_w–q₎≤CM #

A,η

Tb(f)_Lq,ϕ₍_w–q₎

≤C

|α|=m

Dαb_Lip

β(w)wMβ,r,w(f)Lq,ϕ(w–q) =C

|α|=m

Dαb_Lip

β(w)Mβ,r,w(f)Lq,ϕ(w)

≤C

|α|=m

_Dα

b_Lip

β(w) f Lp,β,ϕ(w).

This completes the proof of Theorem .

4 Applications

In this section we shall apply the theorems of the paper to the holomorphic functional calculus of linear elliptic operators. First, we review some deﬁnitions regarding the holo-morphic functional calculus (see []). Given ≤θ<π. Deﬁne

Sθ=

z∈C:arg(z)≤θ∪ {}

and its interior bySθ. SetS˜θ=Sθ\ {}. A closed operatorLon some Banach spaceEis said

to be of typeθif its spectrumσ(L)⊂Sθand for everyν∈(θ,π], there exists a constantCν

such that

|η|(ηI–L)–≤Cν, η∈ ˜/Sθ.

Forν∈(,π], let

H∞Sμ

=f :Sθ→C:f is holomorphic and f L∞<∞

,

where f L∞=sup{|f(z)|:z∈Sμ}. Set

Sμ

=

g∈H∞Sμ

:∃s> ,∃c>  such thatg(z)≤c |z| s

 +|z|s

.

IfLis of typeθandg∈H∞(Sμ), we deﬁneg(L)∈L(E) by

g(L) = –(πi)–

(17)

where is the contour {ξ =re±iφ _: _r_≥__} _{parameterized clockwise around} _S θ with

θ<φ<μ. If, in addition,Lis one-to-one and has a dense range, then, forf ∈H∞(Sμ),

f(L) =h(L)–(fh)(L),

whereh(z) =z( +z)–.Lis said to have a bounded holomorphic functional calculus on the sectorSμif

g(L)≤N g L∞

for someN>  and for allg∈H∞(Sμ).

Now, letLbe a linear operator onL(Rn) withθ<π/ so that (–L) generates a holomor-phic semigroupe–zL, ≤ |arg(z)|<π/ –θ. Applying Theorem  of [] and Theorems -, we get the following.

Corollary Assume that the following conditions are satisﬁed:

(i) The holomorphic semigroupe–zL_,__{≤ |}_arg₍_z₎_|_<_π_{/ –}_θ_{is represented by the kernels} az(x,y)which satisfy,for allν>θ,an upper bound

az(x,y)≤cνh|z|(x,y)

forx,y∈Rn,and≤ |arg(z)|<π/ –θ,whereht(x,y) =Ct–n/s(|x–y|/t)andsis a

positive,bounded and decreasing function satisfying

lim r→∞r

n+

sr= .

(ii) The operatorLhas a bounded holomorphic functional calculus inL₍_Rn₎_;_{that is}_,_for

allν>θ andg∈H∞(Sμ),the operatorg(L)satisﬁes

g(L)(f)_L≤cν g L∞ f L.

Then Theorems-hold for the multilinear operator g(L)b_{associated to g}₍_L₎_{and b}_.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author completed the paper, and read and approved the ﬁnal manuscript.

Acknowledgements

Project was supported by Scientiﬁc Research Fund of Hunan Provincial Education Departments (13C1007).

Received: 18 January 2014 Accepted: 24 May 2014 Published: 24 July 2014

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doi:10.1186/1029-242X-2014-276

Cite this article as:Lu:Weighted boundedness of multilinear operators associated to singular integral operators