Endpoint estimates for vector valued multilinear commutator of fractional area integral operator

(1)

R E S E A R C H

Open Access

Endpoint estimates for vector-valued

multilinear commutator of fractional area

integral operator

Weiping Kuang

*

*_{Correspondence:}

[email protected] Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, P.R. of China

Abstract

In this paper, we prove the endpoint estimates for vector-valued multilinear commutator of fractional area integral operator.

MSC: 42B20; 42B25

Keywords: vector-valued multilinear commutator; Triebel-Lizorkin space; Lipschitz space; Lebesgue space;BMO(Rn₎

1 Introduction

Letb∈BMO(Rn) andTbe the Calderón-Zygmund operator, the commutator [b,T] gen-erated bybandTis deﬁned by

[b,T](f)(x) =b(x)T(f)(x) –T(bf)(x).

A classical result of Coifman, Rochberb and Weiss (see []) proved that the commuta-tor [b,T] is bounded onLp₍_Rn_{) ( <}_p_<_∞_{). In [–], the boundedness properties of the}

commutators for the extreme values ofpare obtained. In this paper, we introduce vector-valued multilinear commutator of fractional area integral operator and prove the endpoint estimates for the commutator|Sbψ,δ|rgenerated by the fractional area integral operatorSψ,δ

andBMOfunctions.

2 Notations and results

We give the following deﬁnitions (see [, , –]).

Deﬁnition  Let  <δ<n, a functionψ satisﬁes:

() _Rnψ(x)dx= ;

() |ψ(x)| ≤C( +|x|)–(n+–δ)_;

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

Suppose that  <r<∞,bj(j= , . . . ,m) are the ﬁxed locally integrable functions onRn_.

Set(x) ={(y,t)∈R₊n+:|x–y| ≤t}and the eigenfunction byχ(x). We deﬁne the

vector-valued multilinear commutator of fractional area integral operator by

S_ψb_,_δ(f)(x)_r=

_∞

i=

Sb_ψ_,_δ(fi)(x)

r

/r

,

(2)

where

Sbψ,δ(f)(x) = (x)

F_tb(f)(x,y)dy dt

tn+

/

and

Fb˜ t(f)(x) =

Rn

_m

j=

bj(x) –bj(z)

ψt(y–z)f(z)dz.

Deﬁnition  We call a locally integrable functionbin the centralBMOspace, namely

CMO(Rn), if the functionbsatisﬁes

bCMO=sup r>

Q(,r)–

Q

b(y) –bQdy<∞.

We have

bCMO≈sup r>

inf c∈C

Q(,r)–

Q

b(y) –cdy.

Deﬁnition  Let  <δ<n,  <p<n/δ. We call a locally integrable functionbinBδ

p(Rn), if

the functionbsatisﬁes

b_Bδ

p=sup

r>

r–n(/p–δ/n)_bχ

Q(,r)Lp<∞.

Now we state our theorems as follows.

Theorem  Suppose <r<∞,  <δ<n,andb= (b, . . . ,bm)for bj∈BMO, ≤j≤m.

Then|Sbψ,δ|ris bounded from Ln/δto BMO(Rn).

Theorem  Let <r<∞,  <δ<n,  <p<n/δ,andb= (b, . . . ,bm),with bj∈BMO(Rn),

for≤j≤m.Then|Sb

ψ,δ|ris bounded from Bδp(Rn)to CMO(Rn).

3 Proofs of theorems

We begin with a preliminaries lemma.

Lemma (see [, ]) Let <r<∞,  <δ<n,  <p<n/δ, /q= /p–δ/n.Then|Sψ,δ|ris bounded from Lp₍_Rn₎_{to L}q₍_Rn_).

Proof of Theorem It is only to prove that there exists a constantCQ, the following in-equality holds:



|Q|

Q

Sb

ψ,δ(f)(x)r–CQdx≤C|f|rLn/δ.

Fix a cubeQ=Q(x,r), letf =g+h={gi}+{hi}forgi=fiχQ,hi=fiχ(Q)c.

Whenm= , set (b)Q=|Q|–

Qb(y)dy, then Ftb(fi)(x,y) =

b(x) – (b)Q

Ft(fi)(y) –Ftb– (b)Q

gi(y) –Ftb– (b)Q

(3)

so

Sψb,δ(f)(x)r–Sψ,δ

(b)Q–b

h(x)r

≤

_∞

i=

χ(x)

b(x) – (b)Q

Ft(fi)(y)r

/r

+

_∞

i=

χ(x)Ft

(b)Q–b

gi

(y)r

/r

+χ(x)Ft

b– (b)Q

f

(y) –χ(x)Ft

b– (b)Q

h(y)_r =A(x) +B(x) +C(x).

ForA(x), suppose  <p<n/δ, /q= /p–δ/nand /q+ /q= , by the Hölder inequality, then



|Q|

Q

_A(x)dx= 

|Q|

Q

_b_(x) – (b)QSψ,δ(f)(x)_rdx

≤ _|

Q|

Q

b(x) – (b)Q q

dx

/q

× _|

Q|

Rn

Sψ,δ(f)(x)

q

rχQ(x)dx

/q

≤CbBMO|Q|–/q Rn

f(x)p_rχQ(x)dx

/p

≤CbBMO|Q|–/q

×

Rn

_f(x)n_r/δdx

δp/n

Q

χQ(x)dx

–δp/n/p

≤CbBMO|Q|–/q|f|r_Ln/δ|Q|(–δp/n)/p

≤CbBMO|f|r_Ln/δ.

ForB(x), ﬁx  <u<n/δ, /v= /u–δ/n, by the Hölder inequality, then 

|Q|

Q

B(x)dx

= 

|Q|

Q

Sψ,δ

b– (b)Q

g(x)_rdx

≤ 

|Q|

Rn

Sψ,δ

b– (b)Q)g

(x)vdx_r

/v

≤C|Q|–/v Rn

b(x) – (b)Q u

f(x)u_rχQ(x)dx

/u

≤C 

|Q|

Q

b(x) – (b)Qudx

/u

_|f|r_Ln/δ

(4)

ForC(x), we have

C(x) =χ(x)Ft

b– (b)Q

f

(y) –χ(x)Ft

b– (b)Q

h(y)_r

≤

Rn+ + Qc

χ(x)–χ(x)b(z) – (b)Qψt(y–z)f(z)_rdz



dy dt tn+

/

≤C

Qc

b(z) – (b)Qf(z)_r

×

|x–y|≤t

t–n_{dy dt}

(t+|y–z|)n+–δ –

|x–y|≤t

t–n_{dy dt}

(t+|y–z|)n+–δ

/dz

≤C

Qc

b(z) – (b)Qf(z)_r

×

|y|≤t,|x+y–z|≤t



t+|x+y–z|n+–δ

– 

t+|x+y–z|

n+–δ

dy dt_tn–

/

dz

≤

Qc

b(z) – (b)Qf(z)_r

|y|≤t,|x+y–z|≤t

|x–x|t–n

(t+|x+y–z|)n+–δ dy dt

/

dz.

Notice that when|y| ≤t, t+|x+y–z| ≥t+|x–z|–|y| ≥t+|x–z|, and

∞



t dt

(t+|x–z|)n+–δ =C|x–z| –n–+δ

,

then, forx∈Q,

C(x)≤

Qc

b(z) – (b)Qf(z)_r

|y|≤t

n+–δ_|_x

–x|t–ndy dt

(t+ |x+y–z|)n+–δ

/

dz

≤C

Qc

b(z) – (b)Qf(z)_r|x–x|/

|y|≤t

t–ndy dt

(t+|x–z|)n+–δ

/

dz

≤C

Qc

b(z) – (b)Qf(z)_r|x–x|/

∞



t dt

(t+|x–z|)n+–δ

/

dz

≤C

Qc

b(z) – (b)Qf(z)_r |

x–x|/ |x–z|n+/–δ

dz

≤C ∞

k=

k+_Q_\_k_Q

b(z) – (b)Qf(z)_r

|x–x|/ |x–z|n+/–δ

dz

≤C ∞

k=

–k/k+Q–+δ/n

k+_Q

b(z) – (b)Qf(z)_rdz

≤CbBMO ∞

k=

k–k/|f|r_Ln/δ

(5)

so that 

|Q|

Q

C(x)dx≤CbBMO|f|r_Ln/δ.

Whenm> , letbQ= ((b)Q, . . . , (bm)Q)∈Rn, where

(bj)Q=|Q|–

Q

bj(y)dy, ≤j≤m,

letf =g+h={gi}+{hi}forgi=fiχQ,hi=fiχ(Q)c. We have

F_tb(fi)(x,y) =

Rn

_m

j=

b(x) –b(z)

ψt(y–z)fi(z)dz

=b(x) – (b)Q

· · ·bm(x) – (bm)Q

Ft(fi)(y) + (–)mFtb– (b)Q

· · ·bm– (bm)Q

fi(y)

+

m–

j=

σ∈Cm_j

(–)m–jb(x) –bQ

σ

×

Rn

b(z) –bQ_σcψt(y–z)fi(z)dz

=b(x) – (b)Q

· · ·bm(x) – (bm)Q

Ft(fi)(y) + (–)mFtb– (b)Q

· · ·bm– (bm)Q

gi(y) + (–)m_F

t

b– (b)Q

· · ·bm– (bm)Q

hi

(y)

+

m–

j=

σ∈Cm_j

(–)m–jb(x) –bQ_σFt(b–bQ)σcfi(x,y),

by the Minkowski inequality, we have

Sbψ,δ(f)(x)_r–Sψ,δ

(b)Q–b

· · ·(bm)Q–bm

h(x)_r ≤χ(x)

b(x) – (b)Q

· · ·bm(x) – (bm)Q

Ft(f)(y)_r

+

m–

j=

σ∈Cm_j

χ(x)

b(x) –bQ_σFt(b–bQ)σcf(x,y)

r

+χ(x)Ft

b– (b)Q

· · ·bm– (bm)Q

g(y)_r

+

χ(x)Ft

_m

j=

bj– (bj)Q

h

(y) –χ(x)Ft

_m

j=

bj– (bj)Q

h

(y)

r

(6)

ForM(x), similar to the proof ofm= , we take  <p<n/δ, /q= /p–δ/n, by the Hölder

inequality and Lemma , we have 

|Q|

Q

M(x)dx

≤



|Q|

Q

m

j=

bj(x) – (bj)Q

q

dx

/q



|Q|

Q

Sψ,δ(f)(x)

q rdx

/q

≤CbBMO|Q|–/q Q

f(x)p_rdx

/p

≤CbBMO|Q|–/q Q

f(x)n_r/δdx

δ/n

|Q|(–(δp/n))/p

≤CbBMO|f|r_Ln/δ.

ForM(x), taking  <p<n/δ, /q= /p–δ/n, we get



|Q|

Q

M(x)dx

≤C m–

j=

σ∈C_jm

bσBMO|Q|–/q Rn

b(x) –bQ

σcf(x)

p

rχQ(x)dx

/p

≤C m–

j=

σ∈Cjm

bσBMO



|Q|

Q

b(x) –bQ_σc

q dx

/q

_|f|r_Ln/δ

≤C m–

j=

σ∈C_jm

bσBMObσc_BMOf_Ln/δ

ForM(x), taking  <p<n/δ, /q= /p–δ/n, we obtain



|Q|

Q

M(x)dx≤



|Q|

Q

Sψ,δ

b– (b)Q

· · ·bm– (bm)Q

g(x)q_rdx

/q

≤C|Q|–/qb(x) – (b)Q

· · ·bm(x) – (bm)Qg(x)_r_Lp ≤C 

|Q|

Q

b(x) – (b)Q

· · ·bm(x) – (bm)Qqdx

/q

_|f|r_Ln/δ

ForM(x), we have

M(x)≤C

Qc|x–x| /_|_x

–z|–(n+/–δ)

m

j=

bj(z) – (bj)Q

f(z)rdz

≤C ∞

k=

k_Q_\_k–_Q|x–x|

/_|_x

–z|–(n+/–δ)

m

j=

bj(z) – (bj)Q

(7)

≤C ∞

k=

k_Q_\_k–_Q

|x–x|/ |x–z|n+/–δ

m

j=

bj(z) – (bj)Q

f(z)rdz

≤C ∞

k=

–k/ 

|k_Q_|–δ/n

k_Q

m

j=

bj(z) – (bj)Q

f(z)rdz

≤C ∞

k=

–k/

k_Q

f(z)n_r/δdz

δ/n

×



|k_Q_|

k_Q

m

j=

bj(z) – (bj)Q

n/(n–δ)

dz

(n–δ)/n

≤C ∞

k=

km–k/

m

j=

bjBMO|f|r_Ln/δ

≤CbBMO|f|r_Ln/δ,

so 

|Q|

Q

M(x)dx≤CbBMO|f|r_Ln/δ.

This completes the proof of Theorem .

Proof of Theorem It is only to prove that there exists a constantCQ, for any of the cubes

Q=Q(,d) (d> ), the following inequality holds: 

|Q|

Q

Sbψ,δ(f)(x)r–CQdx≤CfBδp.

Fix a cubeQ=Q(,d) (d> ). Letf =g+h={gi}+{hi}, wheregi=fiχQ,hi=fiχ(Q)cand

bQ= ((b)Q, . . . , (bm)Q). For (bj)Q=|Q|–

Q|bj(y)|dy, ≤j≤m, we have

F_tb(fi)(x,y) =

Rn

_m

j=

b(x) –b(z)

ψt(y–z)fi(z)dz

=b(x) – (b)Q

· · ·bm(x) – (bm)Q

Ft(fi)(y)

+ (–)mFtb– (b)Q

· · ·bm– (bm)Q

gi(y) + (–)mFtb– (b)Q

· · ·bm– (bm)Q

hi(y) +

m–

j=

σ∈Cm_j

(–)m–jb(x) –bQ_σFt(b–bQ)σcfi

(x,y).

By the Minkowski inequality, we have

Sbψ,δ(f)(x)r–Sψ,δ

(b)Q–b

· · ·(bm)Q–bm

h(x)_r ≤χ(x)

b(x) – (b)Q

· · ·bm(x) – (bm)Q

Ft(f)(y)_r

+

m–

j=

σ∈Cm_j

χ(x)

b(x) –bQ_σFt(b–bQ)σcf(x,y)

(8)

+χ(x)Ft

b– (b)Q

· · ·bm– (bm)Q

g(y)_r +

χ(x)Ft

_m

j=

bj– (bj)Q

h

(y) –χ(x)Ft

_m

j=

bj– (bj)Q

h

(y)

r

=H(x) +H(x) +H(x) +H(x).

ForH(x), take /q= /p–δ/n, by the Hölder inequality and Lemma , we have



|Q|

Q

H(x)dx

≤



|Q|

Q

m

j=

bj(x) – (bj)Q

q

dx

/q



|Q|

Q

Sψ,δ(f)(x)

q rdx

/q

≤CbBMO|Q|–/q Rn

f(x)p_rχQ(x)dx

/p

≤CbBMOd–n(/p–δ/n)|f|rχQ_Lp ≤CbBMO|f|r_Bδ

p.

ForH(x), taking  <u<p<n/δ, /v= /u–δ/n, we get



|Q|

Q

H(x)dx≤C

m–

j=

σ∈Cm_j



|Q|

Q

b(x) –bQ_σvdx

/v

× 

|Q|

Q

Sψ,δ

(b–bQ)σcf

(x)u_rdx

/u

≤C m–

j=

σ∈Cm_j

bσBMO|Q|–/v Rn

b(x) –bQ

σcf(x)

u

rχQ(x)dx

/u

≤C m–

j=

σ∈Cm j

bσBMO|Q|(δ/n–/p)|f|rχQ_Lp

× 

|Q|

Q

b(x) –bQ_σc

pr/(p–r)

dx

(p–u)/pu

≤C m–

j=

σ∈Cm_j

bσBMObσcBMOd–n(/p–δ/n)|f|rχQ_Lp

≤CbBMO|f|r_Bδ

p.

ForH(x), taking  <u<p<n/δ, /v= /u–δ/n, we obtain



|Q|

Q

H(x)dx

≤ _|

Q|

Q

Sψ,δ

b– (b)Q

· · ·bm– (bm)Q

g(x)v_rdx

(9)

≤C|Q|–/v Q

b(x) – (b)Q

· · ·bm(x) – (bm)Q

g(x)u_rdx

/u

≤C|Q|–/vb(x) – (b)Q

· · ·bm(x) – (bm)Q

|f|rχQ_Lv ≤CbBMO|f|r_Bδ

p.

ForH(x), we have

I(x)≤

Rn++

Qc|

χ(x)–χ(x)|

m

j=

bj(z) – (bj)Qψt(y–z)f(z)_rdz



dy dt tn+

/

≤C ∞

k=

k+_Q_\_k_Q|

x–x|/|x–z|–(n+/–δ)

m

j=

bj(z) – (bj)Q

f(z)rdz

≤C ∞

k=

–k/k+Q–+δ/n

k+_Q

m

j=

bj(z) – (bj)Q

f(z)rdz

≤C ∞

k=

km–k/kQ–(/p–δ/n)bBMO|f|rχk_Q

Lp

≤CbBMO|f|r_Bδ

p,

so 

|Q|

Q

H(x)dx≤CbBMO|f|r_Bδ

p.

This completes the proof of Theorem .

Competing interests

The author declares that they have no competing interests.

Received: 1 May 2013 Accepted: 24 September 2013 Published:08 Nov 2013 References

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