Some sharp inequalities for multilinear integral operators

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R E S E A R C H

Open Access

Some sharp inequalities for multilinear

integral operators

Daqing Lu

*

*_{Correspondence:}

[email protected]

College of Mathematics, Changsha University of Science and Technology, Changsha, 410077, P.R. China

Abstract

In this paper, some sharp inequalities for certain multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained. As an application, we obtain the (Lp_,_Lq_{)-norm inequalities and Morrey spaces boundedness} for the multilinear operators.

MSC: 42B20; 42B25

Keywords: multilinear operator; Littlewood-Paley operator; Marcinkiewicz operator; Morrey space; BMO

1 Introduction and results

In this paper, we study some multilinear operators related to some integral operators, whose deﬁnitions are as follows.

Fixn>δ≥. We denote(x) ={(y,t)∈Rn+

+ :|x–y|<t}and the characteristic function of(x) byχ(x). Suppose thatmjare the positive integers (j= , . . . ,l),m+· · ·+ml=m andAjare the functions onRn(j= , . . . ,l). Let

Rmj+(Aj;x,y) =Aj(x) –

|α|≤mj  α!D

α

Aj(y)(x–y)α.

Definition  Letε>  andψbe a fixed function which satisfies the following properties: () _Rnψ(x)dx= ,

() |ψ(x)| ≤C( +|x|)–(n+–δ)_,

() |ψ(x+y) –ψ(x)| ≤C|y|ε_{( +}_|_x_|₎–(n++ε–δ)_when__|_y_|_<_|_x_|_.

The multilinear Littlewood-Paley operator is deﬁned by

SAψ(f)(x) =

(x)

F_tA(f)(x,y)dy dt

tn+

/ ,

where

F_tA(f)(x,y) =

Rn

l

j=Rmj+(Aj;x,z)

|x–z|m ψt(y–z)f(z)dz

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andψt(x) =t–n+δ_ψ(_x_/_t_{) for}_t_{> . Set}_F

t(f)(y) =f∗ψt(y). We also deﬁne that

Sψ(f)(x) =

(x)

Ft(f)(y) dy dt

tn+

/ ,

which is the Littlewood-Paley operator (see []). LetH be the Hilbert spaceH={h:h= ( _Rn+

+ |h(y,t)|

_{dy dt}_/_tn+₎/_<_∞}_{. Then for}

each ﬁxedx∈Rn,F_tA(f)(x,y) may be viewed as a mapping from (, +∞) toH, and it is clear that

SAψ(f)(x) =χ(x)FtA(f)(x,y), Sψ(f)(x) =χ(x)Ft(f)(y).

Deﬁnition  Let  < γ ≤  and be homogeneous of degree zero on Rn _with

Sn– (x)dσ(x) = . Assume that ∈Lipγ(Sn–), that is, there exists a constantM> 

such that for anyx,y∈Sn–_,_|₍_x_{) –}₍_y₎_{| ≤}_M_|_x_–_y_|γ_{. The multilinear Marcinkiewicz}

operator is deﬁned by

μA_S(f)(x) =

(x)

FA t(f)(x,y)

dy dt

tn+

/ ,

where

F_tA(f)(x,y) =

|y–z|≤t

l

j=Rmj+(Aj;x,z)

|x–z|m

(y–z)

|y–z|n––δf(z)dz.

Set

Ft(f)(y) =

|y–z|≤t

(y–z)

|y–z|n––δf(z)dz.

We also deﬁne that

μS(f)(x) =

(x)

Ft(f)(y)dy dt

tn+

/ ,

which is the Marcinkiewicz operator (see []). LetHbe the Hilbert spaceH={h:h= ( _Rn+

+ |h(y,t)|

_{dy dt}_/_tn+₎/_<_∞}_{, then for}

each ﬁxedx∈Rn_, _FA

t (f)(x,y) may be viewed as a mapping from (, +∞) toH, and it is clear that

μA_S(f)(x) =χ(x)FtA(f)(x,y), μS(f)(x) =χ(x)Ft(f)(y). Note that whenm= ,SA

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sharp inequalities for the multilinear integral operatorsSA

ψandμAS whenDαAj∈BMO(Rn) for allα with |α|=mj. As an application, we obtain the (Lp_,_Lq_{)-norm inequalities and} Morrey spaces boundedness for the multilinear operators.

First, 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 function} off is deﬁned by

f#(x) =sup

Q x 

|Q|

Q

f(y) –fQdy,

where, and in what follows,fQ=|Q|–

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

f#(x) =sup

Q x

inf

c∈C 

|Q|

Q

f(y) –cdy.

We say thatf belongs toBMO(Rn) iff#belongs toL∞(Rn) andfBMO=f#L∞. For ≤

p<∞and ≤δ<n, let

Mδ,p(f)(x) =sup Q x



|Q|–pδ/n

Q

f(y)pdy /p

;

we write thatMμ(f) =Mnμ,(f), which is the fractional maximal operator.

Fixedλ> . For ≤p<∞, let

fLp,λ= sup x∈Rn_,d>



dλ

B(x,d)

_f₍_y₎p

dy /p

,

whereB(x,d) ={y∈Rn_:_|_x_–_y_|_<_d_}_{. The Morrey spaces are deﬁned by (see [–])}

Lp,λRn=f ∈L_loc Rn:fLp,λ<∞

.

As the Morrey space may be considered as an extension of the Lebesgue space, it is nat-ural and important to study the boundedness of the multilinear integral operator on the Morrey space.

We shall prove the following theorems.

Theorem  Let Dα_A

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

() Then there exists a constantC> such that for anyf ∈C_∞(Rn_),_{ <}_r_<_n_/δ_and

x∈Rn_,

SAψ(f)

# (x)≤C

l

j=

|αj|=mj

Dαj_A j_BMO

Mδ,r(f)(x);

() If <p<n/δand/p– /q=δ/n,thenSA

ψ is bounded fromLp(Rn)toLq(Rn),that is,

_SA

ψ(f)Lq≤C l

j=

|αj|=mj

_Dαj_A j_BMO

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() If <p<n/δ, <λ<n–pδ,/q= /p–δ/(n–λ),thenSA

ψis bounded fromLp,λ(Rn)

toLq,λ₍_Rn_),_{that is}_,

SAψ(f)_Lq,λ≤C l

j=

|αj|=mj

Dαj_A j_BMO

f_Lp,λ.

Theorem  Let Dα_A

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

() Then there exists a constantC> such that for anyf ∈C_∞(Rn_),_{ <}_r_<_n_/δ_and

x∈Rn_,

μA_S(f)#(x)≤C

l

j=

|αj|=mj

Dαj_A j_BMO

Mδ,r(f)(x);

() If <p<n/δand/p– /q=δ/n,thenμA_S is bounded fromLp(Rn)toLq(Rn),that is,

μA_S(f)_Lq≤C l

j=

|αj|=mj

Dαj_A j_BMO

fLp;

() If <p<n/δ, <λ<n–pδ,/q= /p–δ/(n–λ),thenμA

S is bounded fromLp,λ(Rn)

toLq,λ₍_Rn_),_{that is}_,

μA_S(f)_Lq,λ≤C l

j=

|αj|=mj

Dαj_A j_BMO

f_Lp,λ.

Remark The conclusions of Theorems  and  are completely the same. Thus, they ex-plain that the Littlewood-Paley and Marcinkiewicz operators have the many similar bond-edness properties.

2 Proofs of theorems

To prove the theorems, we need the following lemmas.

Lemma [] Let A be a function on Rn_{and D}α_A_∈_Lq₍_Rn₎_{for all}_α_with_|_α_|₌_{m and some}

q>n.Then

Rm(A;x,y)≤C|x–y|m

|α|=m 

| ˜Q(x,y)|

˜

Q(x,y)

DαA(z)qdz /q

,

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

Lemma [] Suppose that≤r<p<n/δand/q= /p–δ/n.Then

Mδ,r(f)_Lq≤CfLp.

Lemma [, ] Let <p<∞and <λ<n.Then the following estimates hold: (a) M(f)Lp,λ≤Cf#_Lp,λ;

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Lemma  Let <p<n/δand/q= /p–δ/n.Then SψandμSare all bounded from Lp(Rn)

to Lq₍_Rn_).

Proof ForSψ, by Minkowski inequality and the condition ofψ, we have

Sψ(f)(x)≤

Rn

f(z)

(x)

ψt(y–z) dy dt

t+n

/

dz

≤C

Rn

f(z)

∞



|x–y|≤t

t–n+δ

( +|y–z|/t)n+–δ

dy dt t+n

/

dz

≤C

Rn

f(z)

∞



|x–y|≤t

n+t–n

(t+|y–z|)n+–δ dy dt

/

dz,

noting that t+|y–z| ≥t+|x–z|–|x–y| ≥t+|x–z|when|x–y| ≤tand

∞



t dt

(t+|x–z|)n+–δ =C|x–z|

–n+δ

,

we obtain

Sψ(f)(x)≤C

Rn

f(z)

∞



t dt

(t+|x–z|)n+–δ

/

dz

=C

Rn

|f(z)|

|x–z|n–δ dz.

ForμS, note that|x–z| ≤t,|y–z| ≥ |x–z|–t≥ |x–z|– twhen|x–y| ≤t,|y–z| ≤t, we have

μS(f)(x)≤

Rn _|_x–y_|≤_t

| (y–z)f(z)|

|y–z|n––δ



χ(z)(y,t)

dy dt tn+

/

dz

≤C

Rn

f(z)

|x–y|≤t

χ(z)(y,t)t–n–

(|x–z|– t)n––δdy dt

/

dz

≤C

Rn

|f(z)|

|x–z|/

∞

|x–z|/

dt

(|x–z|– t)n–

/

dz

≤C

Rn

|f(z)|

|x–z|n–δ dz.

Thus, the lemma follows from [].

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

, the fol-lowing inequality holds:



|Q|

Q

SAψ(f)(x) –Cdx≤C l

j=

|αj|=mj

Dαj_A j_BMO

Mδ,r(f)(x).

Without loss of generality, we may assumel= . Fix a cubeQ=Q(x,d) andx˜∈Q. Let

˜

Q= √nQandA˜j(x) =Aj(x) –_|α|=mj 

α!(D

α_A_j)

˜

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Dα_A˜

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

F_tA(f)(x,y) =

Rn



j=Rmj+(A˜j;x,z)

|x–z|m ψt(y–z)f(z)dz

=

Rn



j=Rmj+(A˜j;x,z)

|x–z|m ψt(y–z)f(z)dz

+

Rn



j=Rmj(A˜j;x,z)

|x–z|m ψt(y–z)f(z)dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)α

|x–z|m D

α_A˜

(z)ψt(y–z)f(z)dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)α

|x–z|m D

α_A˜_(_z_)ψ

t(y–z)f(z)dz

+

|α|=m,|α|=m  α!α!

Rn

(x–z)α+α_Dα_A˜_(_z₎_Dα_A˜_(_z₎

|x–z|m ψt(y–z)f(z)dz,

then

SψA(f)(x) –S

˜

A

ψ(f)(x)

=χ(x)FtA(f)(x,y)–χ(x)FtA˜(f)(x,y)

≤χ(x)FtA(f)(x,y) –χ(x)F

˜

A

t (f)(x,y)

≤χ(x)

Rn



|x–z|m ψt(y–z)f(z)dz

+χ(x)

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)α

|x–z|m D

α_A˜_(_z_)ψ_t(_y_–_z₎_f_(_z₎_dz

+χ(x)

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)α

|x–z|m D

α_A˜

(z)ψt(y–z)f(z)dz

+χ(x)

|α|=m,|α|=m  α!α!

Rn

(x–z)α+α_Dα_A˜_(_z₎_Dα_A˜_(_z₎

|x–z|m ψt(y–z)f(z)dz

+χ(x)FtA˜(f)(x,y) –χ(x)FtA˜(f)(x,y) :=I(x) +I(x) +I(x) +I(x) +I(x),

thus,



|Q|

Q

SAψ(f)(x) –S

˜

A

ψ(f)(x)dx

≤ 

|Q|

Q

I(x)dx+ C

|Q|

Q

I(x)dx+ C

|Q|

Q

I(x)dx

+ C

|Q|

Q

I(x)dx+ 

|Q|

Q

I(x)dx

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Now, let us estimateI,I,I,IandI, respectively. First, forx∈Qandz∈ ˜Q, by Lemma , we get

Rmj(A˜j;x,z)≤C|x–y| mj

|αj|=mj

_Dαj_A j_BMO.

Thus, by the (Lr,Lq)-boundedness ofSψ, for  <r<n/δand /q= /r–δ/n, we obtain

I≤C



j=

|αj|=mj

Dαj_A j_BMO



|Q|

Q

Sψ(f)(x)dx

≤C



j=

|αj|=mj

Dαj_A j_BMO



|Q|

Q

|Sψ(f)(x)|qdx

/q

≤C



j=

|αj|=mj

Dαj_A j_BMO

|Q|–/q

Q

f(x)rdx /r

≤C



j=

|αj|=mj

Dαj_A j_BMO

Mδ,r(f)(x˜).

ForI, denotingr=pqfor  <p<n/δ,q> , /q+ /q=  and /s= /p–δ/n, we have, by Hölder’s inequality,

I≤C

|α|=m

Dα_A _BMO

|α|=m 

|Q|

Q

Sψ

Dα_A˜

f

(x)dx

≤C

|α|=m

Dα_A _BMO

|α|=m 

|Q|

Rn

Sψ

Dα_A˜

f

(x)sdx /s

≤C

|α|=m

Dα_A _BMO

|α|=m

|Q|–/s

Rn

Dα_A˜_(_x₎_f (x)

p

dx /p

≤C

|α|=m

Dα_A _BMO

×

|α|=m 

|Q|

˜

Q

Dα_A˜_(_x₎pq_dx

/pq 

|Q|–rδ/n

˜

Q

f(x)pqdx /pq

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜).

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

I≤C



j=

|α|=mj

DαAj_BMO

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Similarly, forI, denotingr=pq for  <p<n/δ,q,q,q> , /q+ /q+ /q=  and /s= /p–δ/n, we obtain

I≤C

|α|=m,|α|=m 

|Q|

Q

_S_ψ Dα_A˜

DαA˜f

(x)dx

≤C

|α|=m,|α|=m 

|Q|

Rn

Sψ

Dα_A˜

DαA˜f

(x)sdx /s

≤C

|α|=m,|α|=m

|Q|–/s

Rn

Dα_A˜_(_x₎_Dα_A˜_(_x₎_f_(_x₎p_dx

/p

≤C

|α|=m,|α|=m 

|Q|

˜

Q

Dα_A˜_(_x₎pq_dx

/pq 

|Q|

˜

Q

Dα_A˜_(_x₎pq_dx

/pq

× _| 

Q|–rδ/n

˜

Q

f(x)pqdx /pq

≤C



j=

|α|=mj

Dα_A

j_BMO

Mδ,r(f)(x˜).

ForI, we write

χ(x)FtA˜(f)(x,y) –χ(x)FtA˜(f)(x,y)

=

Rn(χ(x)– χ(x))



|x–z|m ψt(y–z)f(z)dz

+χ(x)

Rn 

|x–z|m – 

|x–z|m



j=

Rmj(A˜j;x,z)ψt(y–z)f(z)dz

+χ(x)

Rn

Rm(A˜;x,z) –Rm(A˜;x,z)

Rm(A˜;x,z)

|x–z|m

ψt(y–z)f(z)dz

+χ(x)

Rn

Rm(A˜;x,z) –Rm(A˜;x,z)

Rm(A˜;x,z)

|x–z|m

ψt(y–z)f(z)dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m –

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m

×Dα_A˜_(_z_)ψ_t(_y_–_z₎_f_(_z₎_dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m –

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m

×Dα_A˜_(_z_)ψt₍_y_–_z₎_f_(_z₎_dz

+

|α|=m,|α|=m  α!α!

Rn

(x–z)α+α_χ

(x)

|x–z|m –

(x–z)α+αχ(x)

|x–z|m

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

|bQ–bQ| ≤Clog

|Q|/|Q|

bBMO forQ⊂Q,

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

Rm(A˜;x,z)≤C|x–z|m

|α|=m

Dα_A

BMO+D

α_A

˜

Q(x,z)–

Dα_A

˜

Q

≤Ck|x–z|m

|α|=m

Dα_A

BMO.

Note that|x–z| ∼ |x–z|forx∈Qandz∈Rn\ ˜Q, we obtain, similar to the proof of Lemma ,

I_()≤

Rn

Rn+ +



j=|Rmj(A˜j;x,z)||ψt(y–z)||f(z)|

|x–z|m

×χ(x)(y,t) –χ(x)(y,t)

_ dy dt

tn+

_/ dz

≤C

Rn



j=|Rmj(A˜j;x,z)||f(z)|

|x–z|m

×

(x)

t–n_{dy dt} (t+|y–z|)n+–δ –

(x)

t–n_{dy dt} (t+|y–z|)n+–δ

/dz

≤C

Rn



j=|Rmj(A˜j;x,z)||f(z)|

|x–z|m

×

|y|≤t

₍_t₊_|_x₊_y_–_z_|₎n+–δ –



(t+|x+y–z|)n+–δ

dy dt_tn–

/

dz

≤C

Rn



j=|Rmj(A˜j;x,z)||f(z)|

|x–z|m

|y|≤t

|x–x|t–ndy dt (t+|x+y–z|)n+–δ

/

dz

≤C

Rn



j=Rmj(A˜j;x,z)|f(z)||x–x|/

|x–z|m+n+/–δ

dz

≤C



j=

|α|=mj

DαAj_BMO

∞

k=

k+_Q_˜_\_k_Q_˜k

 |x–x|/

|x–z|n+/–δ

f(z)dz

≤C



j=

|α|=mj

_Dα

Aj_BMO

∞

k=

k–k/ 

|k_Q˜|–δ/n

k_Q˜

_f₍_z₎_dz

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜);

I_()≤C

Rn

|x–x|

|x–z|m+n+–δ 

j=

Rmj(A˜j;x,z)f(z)dz

≤C



j=

|α|=mj

DαAj_BMO

∞

k=

k+_Q_˜_\_k_Q_˜k

 |x–x|

|x–z|n+–δ

(10)

≤C



j=

|α|=mj

DαAj_BMO

∞

k=

k–k 

|k_Q˜|–δ/n

k_Q˜

f(z)dz

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜).

ForI_()andI_(), by the formula (see [])

Rm(A˜;x,z) –Rm(A˜;x,z) =

|β|<m  β!Rm–|β|

DβA˜;x,x

(x–z)β

and Lemma , we have

Rm(A˜;x,z) –Rm(A˜;x,z)≤C

|β|<m

|α|=m

|x–x|m–|β||x–z||β|DαA_BMO.

Thus, similar to the proof of Lemma ,

I_()≤C



j=

|α|=mj

Dα_A

j_BMO

∞

k=

k+_Q˜_\_k_Q˜

k |x–x| |x–z|n+–δ

f(y)dy

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜);

I_()≤C



j=

|α|=mj

Dα_A

j_BMO

Mδ,r(f)(x˜).

Similarly, we get

I_()≤C

|α|=m

Rn

|x–z|m –

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m

×ψt(y–z)

Dα_A˜

(z)f(z)dz

≤C

|α|=m

DαA_BMO

|α|=m

∞

k=

k–k/+ –k

× 

|k_Q˜_|

k_Q_˜

Dα_A˜ (y)r

dy

/r 

|k_Q˜_|–rδ/n

k_Q_˜

f(y)rdy /r

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜);

_I()

 ≤C



j=

|α|=mj

_Dα

Aj_BMO

(11)

ForI_(), takingq,q>  such that /r+ /q+ /q= , then

I_()≤C

|α|=m,|α|=m

Rn

(x)

|x–z|m –

(x–z)α+αχ(x)

|x–z|m

ψt(y–z)

×Dα_A˜_(_z₎_Dα_A˜_(_z₎_f_(_z₎_dz

≤C

|α|=m,|α|=m

∞

k=

k–k/+ –k 

|k_Q˜|–pδ/n

k_Q˜

f(y)rdy /r

× 

|k_Q˜_|

k_Q˜

Dα_A˜ (y)

q

dy /q



|k_Q˜_|

k_Q˜

Dα_A˜ (y)

q

dy /q

≤C



j=

|α|=mj

_Dα

Aj_BMO

Mδ,r(f)(x˜).

Thus

I ≤C 

j=

|α|=mj

Dα_A

j_BMO

Mδ,r(f)(x˜).

We choose  <r<pin (), then () follows from Lemma . For (), taking  <r<min(p, (n– λ)/pδ) in () and by Lemma , we obtain

S_ψA(f)_Lq,λ≤CM

S_ψA(f)_Lq,λ≤CSAψ(f)

# Lq,λ

≤C



j=

|α|=mj

_Dα

Aj_BMO

Mδ,r(f)_Lq,λ

≤C



j=

|α|=mj

DαAj_BMO

Mrδ/n

|f|r/r_Lq,λ

≤C



j=

|α|=mj

Dα_A

j_BMO

Mrδ/n

|f|r/r_Lq/r,λ

≤C



j=

|α|=mj

DαAj_BMO

|f|r/r_Lp/r,λ

≤C



j=

|α|=mj

DαAj_BMO

f_Lp,λ.

This completes the proof of Theorem .

Proof of Theorem It is only to prove (). LetQ,Q˜,A˜j(x),fandfbe the same as the proof of Theorem . We write

F_tA(f)(x,y)

=

Rn



j=Rmj+(A˜j;x,z)

|x–z|m

(y–z)

(12)

+

Rn



|x–z|m

(y–z)

|y–z|n––δf(z)dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)αDαA˜(z)

|x–z|m

(y–z)

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)αDαA˜(z)

|x–z|m

(y–z)

+

|α|=m,|α|=m  α!α!

Rn

(x–z)α+α_Dα_A˜_(_z₎_Dα_A˜_(_z₎

|x–z|m

(y–z)

|y–z|n––δf(z)dz,

then



|Q|

Q

μA_S(f)(x) –μ_SA˜(f)(x)dx

≤ 

|Q|

Q

χ(x)FtA(f)(x,y) –χ(x)FtA˜(f)(x,y)dx

≤_| Q|

Q

χ(x)

Rn



j=Rmj(A˜j;x,z)

|x–z|m

(y–z)

dx

+ 

|Q|

Q

χ(x)

|α|=m  α!

×

Rn

Rm(A˜;x,z)(x–z)αDαA˜(z)

|x–z|m

(y–z)

dx

+ 

|Q|

Q

χ(x)

|α|=m  α!

×

Rn

Rm(A˜;x,z)(x–z)αDαA˜(z)

|x–z|m

(y–z)

dx

+ 

|Q|

Q

χ(x)

|α|=m,|α|=m  α!α!

×

Rn

(x–z)α+α_Dα_A˜

(z)DαA˜(z)

|x–z|m

(y–z)

dx

+ 

|Q|

Q

χ(x)F

˜

A

t (f)(x,y) –χ(x)F

˜

A

t (f)(x,y)dx

:=J+J+J+J+J.

Similar to the proof of Theorem , we get

J≤C 

j=

|αj|=mj

Dαj_A j_BMO



|Q|

Q

μS(f)(x)dx

≤C



j=

|αj|=mj

Dαj_A j_BMO



|Q|

Q

μS(f)(x)qdx /q

≤C



j=

|αj|=mj

Dαj_A j_BMO

(13)

J≤C

|α|=m

Dα_A _BMO

|α|=m 

|Q|

Q

μS

Dα_A˜

f

(x)dx

≤C

|α|=m

_Dα_A _BMO

|α|=m 

|Q|

Rn

μS

Dα_A˜

f

(x)sdx /s

≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜);

J≤C



j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜);

J≤C

|α|=m,|α|=m 

|Q|

Q

μS

Dα_A˜

DαA˜f

(x)dx

≤C

|α|=m,|α|=m 

|Q|

Rn

μS

Dα_A˜

DαA˜f

(x)sdx /s

≤C



j=

|α|=mj

DαAj_BMO

Mδ,p(f)(x˜).

ForJ, we write

χ(x)FtA˜(f)(x,y) –χ(x)FtA˜(f)(x,y)

=

Rn(χ(x)– χ(x))



j=Rmj(A˜j;x,z)

|x–z|m

(y–z)

|y–z|n––δf(z)dz

+χ(x)

Rn 

|x–z|m – 

|x–z|m



j=

Rmj(A˜j;x,z)

(y–z)

+χ(x)

Rn

Rm(A˜;x,z) –Rm(A˜;x,z)

Rm(A˜;x,z)

|x–z|m

(y–z)

+χ(x)

Rn

Rm(A˜;x,z) –Rm(A˜;x,z)

Rm(A˜;x,z)

|x–z|m

(y–z)

–

|α|=m  α!

Rn

|x–z|m –

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m

×_| (y–z) y–z|n––δD

α_A˜

(z)f(z)dz

–

|α|=m  α!

Rn

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m –

Rm(A˜;x,z)(x–z)αχ(x)

|x–z|m

×_| (y–z) y–z|n––δD

α_A˜_(_z₎_f_(_z₎_dz

+

|α|=m,|α|=m  α!α!

Rn

(x)

|x–z|m –

(x–z)α+αχ(x)

|x–z|m

× (y–z) |y–z|n––δD

(14)

Then, similar to the proof of Lemma  and Theorem , we get

J ≤C 

j=

|α|=mj

DαAj_BMO

Mδ,r(f)(x˜).

The same argument as the proof of Theorem  will give the proof of () and (), we omit

the details and ﬁnish the proof.

Competing interests

The author declares that he has no competing interests.

Received: 21 September 2012 Accepted: 10 May 2013 Published: 4 October 2013 References

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doi:10.1186/1029-242X-2013-445