A sharp inequality for multilinear singular integral operators with non smooth kernels

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R E S E A R C H

Open Access

A sharp inequality for multilinear singular

integral operators with non-smooth kernels

Guangze Gu

*

_{and Mingjie Cai}

*_{Correspondence: [email protected]}

College of Mathematics and Econometrics, Hunan University, Changsha, 410082, P.R. China

Abstract

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted Lp-norm inequality andLlogL-type inequality for the multilinear operators.

MSC: 42B20; 42B25

Keywords: multi-linear operator; singular integral operator with non-smooth kernel; sharp inequality; BMO;Ap-weight

1 Deﬁnitions and results

As the development of singular integral operators and their commutators, multilinear sin-gular integral operators have been well studied (see [–]). In this paper, we study some multilinear operator associated to the singular integral operators with non-smooth ker-nels as follows.

Deﬁnition  A family of operatorsDt,t> , is said to be ‘approximations to the identity’ if, for everyt> ,Dtcan be represented by the kernelat(x,y) in the following sense:

Dt(f)(x) =

Rn

at(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/s|x–y|/t,

wheresis a positive, bounded and decreasing function satisfying

lim

r→∞r n+

sr= 

for some> .

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

T(f)(x) =

RnK(x,y)f(y)dy

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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 thatTBt has an 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 thatAtThas an asso-ciated 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 somec,c> ,δ> .

Letmjbe positive integers (j= , . . . ,l),m+· · ·+ml=m, and letbjbe functions onRn (j= , . . . ,l). Set, for ≤j≤m,

Rmj+(bj;x,y) =bj(x) –

|α|≤mj



α!D

α

bj(y)(x–y)α.

The multilinear operator associated toTis deﬁned by

Tb(f)(x) =

Rn

l

j=Rmj+(bj;x,y)

|x–y|m K(x,y)f(y)dy.

Note that when m= ,Tb _{is just the multilinear commutator of} _T _and_b

j (see []). However, when m> ,Tb is a non-trivial generalization of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [–]). Hu and Yang (see []) proved a variant sharp estimate for the multilinear singular integral operators. In [], Pérez and Trujillo-Gonzalez proved a sharp estimate for the multilinear commutator whenbj∈OscexpLrj(Rn) and noted that Osc_exp_Lrj ⊂BMO. The main purpose of this paper is to prove a sharp

function inequality for the multilinear singular integral operator with non-smooth kernel whenDα_b

j∈BMO(Rn) for allαwith|α|=mj. As an application, we obtain anLp(p> ) norm inequality and an LlogL-type inequality for the multilinear operators. In [–], the boundedness of a singular integral operator with non-smooth kernel is obtained. In [], the boundedness of the commutator associated to the singular integral operator with non-smooth kernel is obtained. Our works are motivated by these papers.

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 function of

f is deﬁned by

f#(x) =sup

Qx 

|Q|

Q

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where, and in what follows,fQ=|Q|–

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

f#₍_x₎_≈_sup

Qxc∈infC



|Q|

Q

f(y) –cdy.

We say thatf belongs toBMO(Rn_{) if}_f#_{belongs to}_L∞₍_Rn_{) and}_f

BMO=f#L∞. LetMbe a Hardy-Littlewood maximal operator deﬁned by

M(f)(x) =sup

Qx 

|Q|

Q

f(y)dy.

Fork∈N, we denote byMk_{the operator}_M_iterated_k_times,_i.e._,_M₍_f₎₍_x_{) =}_M₍_f₎₍_x_{) and}

Mk(f)(x) =MMk–(f)(x) whenk≥.

The sharp maximal functionMA(f) associated with the ‘approximations to the identity’

{At,t> }is deﬁned by

M#_A(f)(x) =sup

Qx 

|Q|

Q

f(y) –AtQ(f)(y)dy,

wheretQ=l(Q)andl(Q) denotes the side length ofQ. For  <r<∞, we denoteM#A(f)r by

M#_A(f)r= M_A#|f|r/r.

Letbe a Young function and˜ be the complementary associated to. For a func-tionf, we denote the-average by

f,Q=inf

λ>  : 

|Q|

Q

_|f

(y)|

λ

dy≤

and the maximal function associated toby

M(f)(x) =sup

Qxf,Q.

The Young functions used in this paper are(t) =t( +logt)r _and˜₍_t_{) =}_exp₍_t/r_{), the} corresponding average and maximal functions are denoted by · L(logL)r_,Q,M_L₍_log_L₎r and · expL/r_,_Q,Mexp_L/r. Following [, , ], we know the generalized Hölder inequality



|Q|

Q

f(y)g(y)dy≤ f,Qg˜,Q

and the following inequality, forr,rj≥,j= , . . . ,lwith /r= /r+· · ·+ /rl, and any

x∈Rn_,_b_∈_BMO₍_Rn_),

f_L₍logL)/r_,_Q≤M_L₍_log_L₎/r(f)≤CM_L₍_log_L₎l(f)≤CMl+(f), b–bQexpLr_,_Q≤Cb_BMO,

|bk+_Q–b__Q| ≤CkbBMO.

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We shall prove the following theorems.

Theorem  If T is a singular integral operator with non-smooth kernel as given in Def-inition,let Dα_b

j∈BMO(Rn)for allα with|α|=mjand j= , . . . ,l.Then there exists a

constant C> such that for any f∈C∞(Rn),  <r< andx˜∈Rn,

M#_ATb(f)_r(x˜)≤C

l

j= |αj|=mj

Dαj_b

j_BMO

Ml+(f)(x˜).

Theorem  If T is a singular integral operator with non-smooth kernel as given in Deﬁni-tion,let Dα_b

j∈BMO(Rn)for allαwith|α|=mjand j= , . . . ,l.Then Tbis bounded on

Lp₍_w₎_{for any}_{ <}_p_<_∞_{and w}_∈_A

p,that is,

Tb(f)_Lp₍_w₎≤C

l

j= |αj|=mj

Dαj_b

j_BMO

fLp₍_w₎.

Theorem  If T is a singular integral operator with non-smooth kernel as given in Deﬁni-tion,let w∈A,Dαbj∈BMO(Rn)for allαwith|α|=mjand j= , . . . ,l.Then there exists

a constant C> such that for allλ> ,

wx∈Rn:Tb(f)(x)>λ≤C

Rn |f(x)|

λ

 +log+

_|f

(x)|

λ

l

w(x)dx.

2 Proof of the theorem

To prove the theorems, we need the following lemma.

Lemma (see []) Let b be a function on Rn_{and D}α_b_∈_Lq₍_Rn₎_{for all}_α_with_|_α_|₌_{m and}

some 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|.

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

/r= /p– /q,

fWLq=sup λ>

λx∈Rn:f(x) >λ/q, Np,q(f) =sup

E f

χELp/χ_E_Lr,

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

fWLq≤Np,q(f)≤

q/(q–p)/pfWLq.

Lemma (see []) Let rj≥for j= , . . . ,l,we denote that/r= /r+· · ·+ /rl.Then



|Q|

Q

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Lemma ([, ]) Let T be a singular integral operator with non-smooth kernel as given in Deﬁnition.Then T is bounded on Lp₍_Rn₎_{for every}_{ <}_p_<_∞_{and bounded from L}₍_Rn₎

to WL₍_Rn_).

Lemma (see [, ]) For anyγ > ,there exists a constant C> independent ofγ such that

x∈Rn:M(f)(x) >Dλ,M_A#(f)(x)≤γ λ≤Cγx∈Rn:M(f)(x) >λ

forλ> ,where D is a ﬁxed constant which only depends on n.Thus

M(f)_Lp≤CM #

A(f)Lp for every f ∈Lp₍_Rn_{),  <}_p_<_∞_.

Lemma  Let{At,t> }be an ‘approximation to the identity’ and b∈BMO(Rn).Then,for

every f ∈Lp₍_Rn_),_p_{> }_and_x_˜_∈_Rn_,

sup

Q˜x 

|Q|

Q

_A_t

Q

(b–bQ)f(x)dx≤CbBMOM(f)(x˜),

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

Proof We write, for any cube Q withx˜∈Q,



|Q|

Q

_A_t

Q

(b–bQ)f

(x)dx≤  |Q|

Q

RnhtQ

(x,y)b(y) –bQ

f(y)dy dx ≤ 

|Q|

Q

Q

htQ(x,y)b(y) –bQ

f(y)dy dx

+ ∞

k= 

|Q|

Q

k+_Q\_k_Q

htQ(x,y)b(y) –bQ

f(y)dy dx

=I+I.

We have, by the generalized Hölder inequality,

I≤C 

|Q||Q|

Q

Q

b(y) –bQ

f(y)dy dx ≤Cb–bQexpL,QfL(logL),Q

≤CbBMOM(f)(x˜).

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

s((k–)₎ |Q| , then

I≤C ∞

k=

s(k–) 

|Q|

Q

k+_Q

b(y) –bQ

f(y)dy dx

≤C

∞

k=

kns(k–) 

|k+Q|

k+_Q

b(y) –bQ

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≤C

∞

k=

kns(k–)b–bQexpL,k+_Qf_L₍_log_L_),k+_Q

≤C

∞

k=

(k–)ns(k–)bBMOM(f)(x˜)

≤CbBMOM(f)(x˜),

where the last inequality follows from

∞

k=

(k–)ns(k–)≤C

∞

k=

–(k–)ε<∞

for some> . This completes the proof.

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

 that the following inequality holds:



|Q|

Q

Tb(f)(x)r–AtQT

b₍_f₎₍_x₎r_dx

/r

≤C

l

j= |αj|=mj

Dαj_b

j_BMO

Ml+(f)(x).

Without loss of generality, we may assumel= . Fix a cubeQ=Q(x,d) andx˜∈Q. LetQ˜ = √nQandb˜j(x) =bj(x) –

|α|=mj



α!(D

α_b

j)Q˜xα, thenRmj(bj;x,y) =Rmj(b˜j;x,y) andD α_b˜

j=

Dα_b

j– (Dαbj)Q˜ for|α|=mj. We write, forf=fχQ˜ +fχRn_{\ ˜}_Q=f+f,

Tb(f)(x) =

Rn



j=Rmj+(b˜j;x,y)

|x–y|m K(x,y)f(y)dy=

Rn



j=Rmj(b˜j;x,y)

|x–y|m K(x,y)f(y)dy

–

|α|=m 

α!

Rn

Rm(b˜;x,y)(x–y)αDαb˜(y)

–

|α|=m 

α!

Rn

Rm(b˜;x,y)(x–y)αDαb˜(y)

+

|α|=m,|α|=m 

α!α!

Rn

(x–y)α+α_Dα_b˜

(y)Dαb˜(y)

+

Rn



j=Rmj+(b˜j;x,y)

|x–y|m K(x,y)f(y)dy

=T



j=Rmj(b˜j;x,·) |x–·|m f

–T

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

–T

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

+T

|α|=m,|α|=m 

α!α!

(x–·)α+α_Dα_b˜ Dαb˜

|x–·|m f

+T



j=Rmj+(b˜j;x,·) |x–·|m f

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and

AtQT

b₍_f₎₍_x_{) =}

Rn



j=Rmj(b˜j;x,y)

|x–y|m Kt(x,y)f(y)dy

–

|α|=m 

α!

Rn

Rm(b˜;x,y)(x–y)αDαb˜(y)

|x–y|m Kt(x,y)f(y)dy

–

|α|=m 

α!

Rn

Rm(b˜;x,y)(x–y)αDαb˜(y)

|x–y|m Kt(x,y)f(y)dy

+

|α|=m,|α|=m 

α!α!

Rn

(x–y)α+α_Dα_b˜

(y)Dαb˜(y)

|x–y|m Kt(x,y)f(y)dy

+

Rn



j=Rmj+(b˜j;x,y)

|x–y|m Kt(x,y)f(y)dy

=AtQT



j=Rmj(b˜j;x,·) |x–·|m f

–AtQT

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

–AtQT

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

+AtQT

|α|=m,|α|=m 

α!α!

(x–·)α+α_Dα_b˜ Dαb˜

|x–·|m f

+AtQT



j=Rmj+(b˜j;x,·) |x–·|m f

,

then



|Q|

Q

Tb(f)(x)r–AtQT

b₍_f₎₍_x₎r

dx /r ≤  |Q| Q

_Tb₍_f₎₍_x_{) –}_A tQT

b₍_f₎₍_x₎r_dx

/r ≤ C |Q| Q T 

j=Rmj(b˜j;x,·) |x–·|m f

rdx /r + C |Q| Q T

|α|=m

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

rdx /r + C |Q| Q T

|α|=m

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

rdx /r + C |Q|

Q

T

|α|=m,|α|=m

Q

(x–·)α+α_Dα_b˜ Dαb˜

|x–·|m f

rdx /r + C |Q| Q AtQT



j=Rmj(b˜j;x,·) |x–·|m f

rdx

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+ C |Q| Q AtQT

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

rdx /r + C |Q|

Q

AtQT

|α|=m 

α!

Rm(b˜;x,·)(x–·)αDαb˜

|x–·|m f

rdx /r + C |Q|

Q

AtQT

|α|=m,|α|=m 

α!α!

(x–·)α+α_Dα_b˜ Dαb˜

|x–·|m f

rdx /r + C |Q|

Q

(T–AtQT)



j=Rmj+(b˜j;x,·) |x–·|m f

rdx

/r

:=I+I+I+I+I+I+I+I+I.

Now, let us estimateI,I,I,I,I,I,I,IandI, respectively. First, forx∈Qandy∈ ˜Q, by Lemma , we get

Rm(b˜j;x,y)≤C|x–y|m |αj|=m

Dαj_b

j_BMO,

by Lemma  and the weak type (, ) ofT(Lemma ), we obtain

I≤C 

j= |αj|=mj

Dαj_b

j_BMO



|Q|

Rn

T(f)(x) r dx /r ≤C 

j= |αj|=mj

Dαj_b

j_BMO

|Q|–T(f)χQLr |Q|/r–

≤C



j= |αj|=mj

Dαj_b

j_BMO

|Q|–_T₍_f )_WL

≤C



j= |αj|=mj

Dαj_b

j_BMO

| ˜Q|–_f L

≤C



j= |αj|=mj

Dαj_b

j_BMO

M(f)(x˜).

ForI, we get, by Lemma  and the generalized Hölder inequality,

I≤C

|α|=m

_Dα_b _BMO

|α|=m

 |Q| Rn _T

Dα_b˜ f

(x)rdx

/r

≤C

|α|=m

Dα_b _BMO

|α|=m

|Q|–T(D α_b˜

f)χQLr |Q|/r–

≤C

|α|=m

Dα_b _BMO

|α|=m

|Q|–TDα_b˜ f_WL

≤C

|α|=m

Dα_b _BMO

|α|=m

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≤C

|α|=m

Dα_b _BMO

|α|=m

Dα_b –

Dαb

˜

QexpL,Q˜fL(logL),Q˜

≤C



j= |αj|=mj

Dαj_b

j_BMO

M(f)(x˜).

ForI, similar to the proof ofI, we get

I≤C 

j= |α|=mj

Dαbj_BMO

M(f)(x˜).

Similarly, forI, takingr,r,r≥ such that /r= /r+ /r, we obtain, by Lemma  and the generalized Hölder inequality,

I≤C

|α|=m,|α|=m



|Q|

Rn

TDα_b˜ Dαb˜f

(x)rdx

/r

≤C

|α|=m,|α|=m

|Q|–T(Dαb˜Dαb˜f)χQLr |Q|/r–

≤C

|α|=m,|α|=m

|Q|–_T_Dα_b˜

Dαb˜f_WL

≤C

|α|=m,|α|=m

|Q|–_Dα_b˜

Dαb˜f_L

≤C

|α|=m,|α|=m 

j=

Dαj_b

j–

Dαj_b

j

˜

QexpLrj,Q˜ · fL(logL)/r_,_Q_˜

≤C



j= |α|=mj

Dαbj_BMO

M(f)(x˜).

ForI,I,IandI, by Lemma , we get

I+I+I+I≤C 

j= |αj|=mj

Dαj_b

j_BMO



|Q|

Q

AtQT(f)(x)dx

+C

|α|=m

Dα_b _BMO

|α|=m 

|Q|

Q

AtQT

Dα_b˜ f

(x)dx

+C

|α|=m

Dα_b _BMO

|α|=m 

|Q|

Q

AtQT

Dα_b˜ f

(x)dx

+C

|α|=m,|α|=m 

|Q|

Q

AtQT

Dα_b˜ Dαb˜f

(x)dx

≤C



j= |α|=mj

Dαbj_BMO

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ForI, we write

(T–AtQT)



j=Rmj+(b˜j;x,·) |x–·|m f

=

Rn



j=Rmj+(b˜j;x,y) |x–y|m

K(x,y) –Kt(x,y)

f(y)dy

=

Rn



j=Rmj(b˜j;x,y) |x–y|m

K(x,y) –Kt(x,y)f(y)dy

–

|α|=m 

α!

Rn Dα_b˜

(y)(x–y)α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Rm(b˜;x,y)

|x–y|m

K(x,y) –Kt(x,y)

f(y)dy

+

|α|=m,|α|=m 

α!α!

Rn Dα_b˜

(y)Dαb˜(y)(x–y)α+α

|x–y|m

K(x,y) –Kt(x,y)

f(y)dy

=I_()+I_()+I_()+I_().

By Lemma  and the following inequality (see [])

|bQ–bQ| ≤Clog

|Q|/|Q|

bBMO forQ⊂Q,

we know that forx∈Qandy∈k+_Q˜_\_k_Q˜_,

Rm(b˜;x,y)≤C|x–y|m

|α|=m

Dα_b

BMO+D

α_b

˜ Q(x,y)–

Dα_b

˜

Q

≤Ck|x–y|m

|α|=m

Dα_b

BMO.

Note that|x–y| ≥d=t/_and_|x_–_{y| ∼ |x}

–y|forx∈Qandy∈Rn\ ˜Q. By the conditions onKandKt, we obtain

I_()= ∞

k=

k+_Q\˜ _k_Q˜



j=|Rmj(b˜j;x,y)|

|x–y|m K(x,y) –Kt(x,y)f(y)dy

≤C



j= |α|=mj

Dαbj_BMO

∞

k=

k+_Q\_˜ _k_Q_˜k

 dδ

|x–y|n+δ

f(y)dy

≤C



j= |α|=mj

Dαbj_BMO

∞

k=

k–δk 

|kQ|˜

k_Q˜

f(y)dy

≤C



j= |α|=mj

Dαbj_BMO

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ForI_(), we get, by the generalized Hölder inequality,

I_()≤C

|α|=m

Dα_b _BMO

|α|=m ∞

k=

k+_Q\_˜ _k_Q_˜ kdδ |x–y|n+δ

Dα_b˜

(y)f(y)dy

≤C

|α|=m

Dα_b _BMO

×

|α|=m ∞

k=

k–δkDα_b –

Dα_b 

˜

QexpL,k_Q˜fL(logL),k_Q_˜

≤C



j= |α|=mj

Dα_b

j_BMO

M(f)(x˜).

Similarly,

I_()≤C



j= |α|=mj

Dαbj_BMO

M(f)(x˜).

ForI(), taking r,r,r≥ such that /r= /r+ /r, by Lemma  and the generalized Hölder inequality, we get

I_()≤C

|α|=m,|α|=m ∞

k=

k+_Q\˜ _k_Q˜ dδ |x–y|n+δ

Dα_b˜

(y)Dαb˜(y)f(y)dy

≤C

|α|=m,|α|=m ∞

k= 

j=

Dαj_b

j–

Dαj_b

j

˜

QexpLrj,k_Q_˜f_L₍logL)/r_,k_Q˜

≤C



j= |α|=mj

Dα_b

j_BMO

M(f)(x˜).

Thus

|I| ≤C 

j= |α|=mj

Dαbj_BMO

M(f)(x˜).

This completes the proof of Theorem .

By Theorem  and theLp₍_w_{)-boundedness of}_Ml+_{, we may obtain the conclusions of} Theorem . By Theorem  and [, ], we may obtain the conclusions of Theorem .

3 Applications

In this section we shall apply Theorems ,  and  of the paper to the holomorphic func-tional calculus of linear elliptic operators. First, we review some deﬁnitions regarding the holomorphic functional calculus (see []). Given ≤θ<π, deﬁne

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

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

Cνsuch that

|η|(ηI–L)–≤Cν, η∈ ˜/Sθ.

Forν∈(,π], let

H∞Sμ

=f :Sθ→C:f is holomorphic andfL∞<∞

,

wherefL∞=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)–

(ηI–L)–g(η)dη,

where is the contour {ξ =re±iφ _: _r_≥__} _{parameterized clockwise around} _Sθ _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)≤NgL∞

for someN>  and for allg∈H∞(Sμ).

Now, letLbe a linear operator onL(Rn) withθ <π/ so that (–L) generates a holo-morphic semigroupe–zL_{, }_{≤ |}_arg₍_z₎_|_<_π_{/ –}_θ_{. Applying Theorem  of [], we get the} following.

Theorem  Assume the following conditions are satisﬁed:

(i)The holomorphic semigroup e–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)

for x,y∈Rn_,_and__{≤ |}_arg₍_z₎_|_<_π_{/ –}_θ_,_{where h}

t(x,y) =Ct–n/s(|x–y|/t)and s is a positive,

bounded and decreasing function satisfying

lim

r→∞r n+

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(ii)The operator L has a bounded holomorphic functional calculus in L₍_Rn_),_{that is}_,_for

allν>θand g∈H∞(Sμ),the operator g(L)satisﬁes

g(L)(f)_L≤cνgL∞fL.

Then,for Dα_b

j∈BMO(Rn)for allαwith|α|=mjand j= , . . . ,l,the multilinear operator

g(L)b_{associated to g}₍_L₎_{and b}

jsatisﬁes: (a) For <r< andx˜∈Rn_,

M#_Ag(L)b(f)_r(x˜)≤C

l

j= |αj|=mj

Dαj_b

j_BMO

Ml+(f)(x˜);

(b) g(L)b_{is bounded on}_Lp₍_w₎_{for any}_{ <}_p_<_∞_and_w_∈_A

p,that is,

g(L)b(f)_Lp₍_w₎≤C

l

j= |αj|=mj

Dαj_b

j_BMO

fLp₍_w₎;

(c) There exists a constantC> such that for allλ> andw∈A,

wx∈Rn:g(L)b(f)(x)>λ≤C

Rn |f(x)|

λ

 +log+

_|f

(x)|

λ

l

w(x)dx.

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: 17 March 2013 Accepted: 12 August 2013 Published: 16 September 2013

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doi:10.1186/1029-242X-2013-439