Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz Hardy spaces

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R E S E A R C H

Open Access

Boundedness for multilinear commutator of

Marcinkiewicz operator with variable kernels

on Hardy and Herz-Hardy spaces

Wenxin Yu

1*

_{, Yigang He}

1,2

_{, Qiwu Luo}

1

_{, Jian Zheng}

1

_{and Xianming Wu}

1

*_{Correspondence:}

[email protected] 1_{College of Electrical and}

Information Engineering, Hunan University, Changsha, Hunan 410082, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, the (Hp

b,L

p_{)- and (}_H_K˙α,p q,b,K˙

α,p

q )-type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels is obtained.

MSC: 42B20; 42B25

Keywords: Marcinkiewicz operator; multilinear commutator; BMO; Hardy space; Herz-Hardy space

1 Introduction and deﬁnitions

LetTbe the Calderón-Zygmund operator andb∈BMO(Rn_{). The commutator [}_b_,_T_]

gen-erated byT andbis deﬁned by

[b,T](f)(x) =b(x)T(f)(x) –T(bf)(x).

A classical result of Coifman, Rochberg and Weiss (see [, ]) proved that the commutator [b,T] is bounded onLp₍_Rn_{) ( <}_p_<_∞_{). However, it was observed that the [}_b_,_T_{] is not}

bounded, in general, fromHp₍_Rn_{) to}_Lp₍_Rn_{). But if}_Hp₍_Rn_{) is replaced by a suitable atomic}

spaceHP

b(Rn), then [b,T] maps continuouslyHbP(Rn) intoLp(Rn) (see []). In addition, we

easily know thatH_bp(Rn)⊂Hp(Rn). In recent years, the theory of Herz-type Hardy spaces have been developed (see [–]). The main purpose of this paper is to consider the conti-nuity of multilinear commutators related to Marcinkiewicz operators with variable kernels andBMO(Rn) functions on certain Hardy and Herz-Hardy spaces. Let us ﬁrst introduce some deﬁnitions (see [–]).

Given a positive integermand ≤j≤m, we denote byCm

j the family of all ﬁnite subsets

σ={σ(), . . . ,σ(j)}of{, . . . ,m}ofjdiﬀerent elements. Forσ∈C_jm, setσc={, . . . ,m} \σ. Forb= (b, . . . ,bm) andσ={σ(), . . . ,σ(j)} ∈Cjm, setbσ = (bσ(), . . . ,bσ(j)),bσ=bσ()· · ·bσ(j) andbσBMO=bσ()BMO· · · bσ(j)BMO.

Deﬁnition  Letbi(i= , . . . ,m) be a locally integrable function and  <p≤. A bounded

measurable functionaonRn_{is said to be a (}_p_,_b_{) atom if} () suppa⊂B=B(x,r),

() aL∞≤ |B|–/p,

() _Ba(y)dy=_Ba(y)_l∈_σbl(y)dy= for anyσ∈Cjm,≤j≤m.

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A temperate distributionf is said to belong toHp

b(R

n_{) if, in the Schwartz distribution}

sense, it can be written as

f(x) =

∞

j= λjaj(x),

where every aj is (p,b) atom, λ ∈ C and

∞

j=|λ|p < ∞. Moreover, fHp b(R

n₎ ≈

(∞_j_=|λj|p)/p.

Given a setE⊂Rn_{, the characteristic function of}_E_{is deﬁned by}_χ

E. LetBk={x∈Rn: |x| ≤k_}_and_C

k=Bk\Bk–andχk=χBk,k∈Z.

Deﬁnition  Let  <p,q<∞,α∈R. Fork∈Z, setBk={x∈Rn:|x| ≤k}andCk=Bk\

Bk–. Denote byχkthe characteristic function ofCkand byχthe characteristic function ofB.

() The homogeneous Herz space is deﬁned by

˙

K_qα,pRn=f ∈Lq_locRn\ {}:fK˙qα,p<∞ ,

where

fK˙qα,p=

_∞

k=–∞

kαpfχkpLq /p

.

() The nonhomogeneous Herz space is deﬁned by

Kα,p q

Rn=f ∈Lq_locRn:f_Kα,p q <∞ ,

where

f_Kα,p

q =

_∞

k=

kαp_f_χ

kp_Lq+fχp_Lq /p

.

Deﬁnition  Letα∈R,  <q<∞,  <α<n( – /q),bi∈BMO(Rn), ≤i≤m. A function

aonRnis called a central (α,q,b)-atom (or a central (α,q,b)-atom of restrict type) if

() suppa⊂B=B(,r)(or for somer≥), () aLq≤ |B|–α/n,

() _Ba(x)dx=_Ba(x)_l∈_σbl(x)dx= for anyσ∈Cjm,≤j≤m.

A temperate distributionf is said to belong toHK˙α,p

q,b(R

n_{) (or}_HKα,p q,b(R

n_{)) if it can be}

written asf =∞_j_=–_∞λjaj(orf =

∞

j=λjaj), in the Schwartz distribution sense, whereaj

is a central (α,q,b)-atom (or a central (α,q,b)-atom of restrict type) supported onB(, j₎

and∞_–_∞|λj|p<∞(or∞j=|λj|p<∞). Moreover,f_HK˙α,p q,b

(orf_HKα,p q,b

)≈(_j|λj|p)/p.

Deﬁnition  Letbe homogeneous of degree zero onRn_{such that}_ω

r(δ) is deﬁned

ωr(δ) =sup |ρ|<δ

Sn–

(ρx´) –(x´)rdσ(x´)

/r

,

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If_ωr(δ)

δ dδ<∞, we say(x) satisﬁedLr-Dinicondition.

Deﬁnition  Let  <<n,  <γ ≤ andbe homogeneous of degree zero onRn_such

that_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 Marcinkiewicz multilinear}

commutator is deﬁned by

μb˜(f)(x) =

∞



F_tb˜(f)(x)dt

t

/ ,

where

F_tb˜(f)(x) =

|x–y|≤t

(x–y)

|x–y|n––

_m

j=

bj(x) –bj(y)

f(y)dy.

Set

Ft(f)(x) =

|x–y|≤t

(x–y)

|x–y|n––f(y)dy,

we also deﬁne that

μ(f)(x) =

∞



_F_t(f)(x)dt

t

/ ,

which is the Marcinkiewicz operator (see [, , ]).

2 Theorems and proofs

We begin with three preliminary lemmas.

Lemma (see []) Let <r<∞,bj∈BMO(Rn)for j= , . . . ,k and k∈N.Then we have



|Q|

Q k

j=

bj(y) – (bj)Qdy≤C k

j=

bjBMO(Rn₎

and



|Q|

Q k

j=

bj(y) – (bj)Q r

dy

/r ≤C

k

j=

bjBMO(Rn₎.

Lemma (see []) Let <<n,  <s<n/and/r= /s–/n.Thenμbis bounded from Ls₍_Rn₎_{to L}r₍_Rn_).

Lemma (see []) Let <μ<n,(x,z)∈L∞(Rn₎_{satisfy L}r₍_Sn–_{) (}_r_≥_)_conditions_,_that

is,there exists a constant <a< /such that for|y|<aR,

R<|x|<R

_|_x(x_–,x_y_|–n–yμ)–

(x,x)

|x–y|n–μ rdx

/r

≤CRn/r–(n–μ)

_|

y| R +

|y|/R

|y|/R

ωr(δ)

δ dδ

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Theorem  Let <<n,n/(n+ / –) <q≤, /q= /p–/n,b= (b, . . . ,bm),bi∈BMO,

≤i≤m.Thenμb

is bounded from H p b(R

n₎_{to L}q₍_Rn_).

Proof It suﬃces to show that there exists a constantC>  such that for every (p,b) atoma,

μb(a)_Lq≤C.

Letabe a (p,b) atom supported on a ballB=B(x, d). We write

Rn

μb(a)(x)qdx=

|x–x|≤d

μb(a)(x)qdx+

|x–x|>d

μb(a)(x)qdx=I+II.

ForI, takingr,s>  with q<s<n/ and /r= /s–/n, by Hölder’s inequality and the (Ls_,_Lr_{)-boundedness of}_μb

, we get

I≤Cμb(a)q_LrB(x, d)–q/r≤Caq_Ls|B|–q/r≤C|B|–q/p+q/s+–q/r≤C.

ForII, denotingλ= (λ, . . . ,λm) withλi= (bi)B, ≤i≤m, where (bi)B=|B(x, d)|–×

B(x,d)bi(x)dx, by Hölder’s inequality and the vanishing moment ofa, we get

II≤

|x–x|+d



|x–y|<t m

j=

bj(x) –bj(y)

a(y)(x,x–y)

|x–y|n–– dy



dt t

/

+

∞

|x–x|+d

|x–y|<t m

j=

bj(x) –bj(y)

a(y)(x,x–y)



dt t

/

=II+II.

Note that|x–y| ∼ |x–x| ∼ |x–x|+ dfor|x–x|> d,y∈B. For /t+ /r= , we have

II≤C

Rn

|x–x|+d

|x–y|

dt t

/m

j=

bj(x) –bj(y)a(y)|

(x,x–y)|

≤C

Rn

_|_x_–_y_| –

 (|x–x|+ d)

/m

j=

bj(x) –bj(y)a(y)|

(x,x–y)|

≤C

Rn m

j=

bj(x) –bj(y)a(y)|

(x,x–y)|

|x–y|n––ε

|y–x|/

|x–x|/

dy

≤C

Rn m

j=

bj(x) –bj(y)a(y) |

(x,x–y)|

|x–x|n+/–|

y–x|/dy

≤C

m–

j=

σ∈Cm j



|x–x|n+/–

B

b(y) –λ_σca(y)(x,x–y)|y–x|/dy

×b(x) –λ_σ

≤C

m–

j=

σ∈C_jm



|x–x|n+/–

B

b(y) –λ_σca(y)|y–x|/

t

dy

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× B

(x,x–y)rdy

/r

b(x) –λ_σ

≤C

m–

j=

σ∈C_jm

dn(–/p+/t+/r+/n)

|x–x|n+/–

bσcBMO· L∞×Lrb(x) –_λ

σ

≤C

m–

j=

σ∈C_jm

dn(–/p+/t+/r+/n)

|x–x|n+/–

bσc_BMOb(x) –λ

σ,

so we have

∞

k=

k+_d≥|x_–_x__|_>k_d

m–

j=

σ∈Cm j

dn(–/p+/t+/r+/n)

|x–x|n+/–

bσc_BMOb(x) –λ

σ q

dx

≤C

m–

j=

σ∈C_jm bσcq

BMO ∞

k=

Ck

dn(–/p+/n) |x–y|n+/–

q

b(x) –λ_σqdx

≤C

m–

j=

σ∈C_jm

bσcq_BMO ∞

k=

dqn(+/n–/q–δ/n)

|k_d|q(n+/–)

kdn

×



|k_B_|

k_B

b(x) –λ_σdx

q

≤C

∞

k=

kq–k(q(n+/–)–n)_bq BMO

≤Cbq_BMO.

ForII, we can obtain

II =

Rn

∞

|x–x|+d

dt t

/

m

j=

bj(x) –bj(y)

a(y)(x,x–y)

≤C

Rn m

j=

bj(x) –bj(y)

a(y)(x,x–y)

|x–y|n–– –

(x,x)

|x|n–– dy



|x–x|+ d

≤C

Rn m

j=

_b_j₍_x_{) –}_b_j₍_y₎_a₍_y₎(x,x–y)

|x–y|n– –

(x,x)

|x|n– dy

≤C

m–

j=

σ∈C_jm

b(x) –λ_σ

B

b(y) –λ_σca(y)

_|_x(x_–,x_y_|–n–y)–

(x,x)

|x|n– dy.

Thus, by Lemma , we have

_∞

k=

k+_d≥|x_–_x |>kd

|II|qdx

/q

≤C

_m_–

j=

σ∈C_jm ∞

k=

Ck

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×

B

(x,x–y)

|x–y|n– –

(x,x)

|x|n– dy

q

dx

/q

≤C

m–

j=

σ∈C_jm ∞

k=

B

×

Ck

b(x) –λ_σq(x,x–y)

|x–y|n– –

(x,x)

|x|n– qdx

/q

dy

≤C

m–

j=

σ∈C_jm ∞

k=

k+B/q–/rbσBMO

×

B

_b₍_y_{) –}λ_σca(y)

Ck

(x,x–y)

|x–y|n– –

(x,x)

|x|n– rdx

/r

dy

≤C

m–

j=

σ∈C_jm ∞

k=

k+dn(/q–/r)kdn/r–(n–)bσBMO

B

×

_|

y| |k_d|+

|y|/|k+_d|_<_δ_<_|y|_/_|_k_d|

ωr(δ)

δ dδ

dy

≤C

m–

j=

σ∈C_jm ∞

k=

kdn/q–n/rkdn/r–(n–+/)bσBMO

×

B

b(y) –λ_σca(y)dy

≤C

∞

k=

|B|kdn/q–(n–+/)bσBMObσc_BMO

≤CbBMO.

This ﬁnishes the proof of Theorem .

Theorem  Let <<n,  <p<∞,  <q,q<∞, /q– /q=/n,n( – /q)≤α<

n( – /q) + / + and bi∈BMO(Rn), ≤i≤m,b= (b, . . . ,bm).Thenμb is bounded from HK˙α,p

q,b(R

n₎_to_K˙α,p q (Rn).

Proof Letf ∈HK˙α,p

q,b(R

n_{) and}_f₍_x_{) =}∞

j=–∞λjaj(x) be the atomic decomposition forf as in

Deﬁnition . We write

μb(f)(x)p_K_˙α,p q_

≤ ∞

k=–∞

kαp _k_–

j=–∞

|λj|μ b

(aj)(x)χk_Lq p

+

∞

k=–∞ kαp

_∞

j=k–

|λj|μb(aj)(x)χk_Lq p

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ForJJ, by the (Lq_,_Lq_{)-boundedness of}_μb

, we get

JJ≤C

∞

k=–∞

kαp

_∞

j=k–

|λj|ajLq p

≤C

∞

k=–∞

kαp

_∞

j=k–

|λj|–jα p

≤ ⎧ ⎨ ⎩

C∞_k_=–_∞kαp₍∞

j=k–|λj|p–jαp),  <p≤,

C∞_k_=–_∞kαp₍∞

j=k–|λj|p–jαp/)(

∞

j=k––jαp _/

)p/p_, _p_{> ,}

≤ ⎧ ⎨ ⎩

C∞_j_=–_∞|λj|p( j+

k=–∞(k–j)αp),  <p≤,

C∞_j_=–_∞|λj|p( j+

k=–∞(k–j)αp/)( j+

k=–∞(k–j)αp

_/

)p/p_, _p_{> ,}

≤C

∞

j=–∞

|λj|p≤Cfp_H_K_˙α,p q,b

.

ForJ, letx∈Bk\Bk–,bij=|Bj|–

Bjbi(x)dx, ≤i≤m,b _{= (}_b

j, . . . ,bmj ), we have

μb(aj)(x) =

∞ 

|x–y|<t m

i=

bi(x) –bi(y)

(x,x–y)

|x–y|n––aj(y)dy  dt t / =

|x|+j



|x–y|<t m

i=

bi(x) –bi(y)

(x,x–y)

|x–y|n––aj(y)dy  dt t / + ∞

|x|+j

|x–y|<t m

i=

bi(x) –bi(y)

(x,x–y)

|x–y|n––aj(y)dy  dt t /

=G+H.

ForG, noting thaty∈Bj,x∈B(, k)\B(, k–),j≤k– , we know|x–y| ∼ |x| ∼ |x|+ j.

Then, similar to the proof of Theorem , we obtain

G≤C

Bj |x|+

j

|x–y|

dt t

/m

i=

bi(x) –bi(y)|

(x,x–y)||aj(y)| |x–y|n–– dy

≤

Bj

_|_x_–_y_| –  (|x|+ j₎

/m

i=

bi(x) –bi(y)

|(x,x–y)||aj(y)| |x–y|n–– dy

≤Cj(/+)

Bj



|x|n+/

m

i=

bi(x) –bi(y)(x,x–y)aj(y)dy

≤C

j(/+)

|x|n+/

m

i=

σ∈Cm i

b(x) –b_σ

Bj

(x,x–y)aj(y)b(y) –b

σcdy

≤C

j(/+)

|x|n+/

m–

j=

σ∈C_jm

Bj

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× B

(x,x–y)rdy

/r

b(x) –b_σ

≤C

j(/++n(/t+/r–/q–α))

|x|n+/

m

i=

σ∈Cm i

bσc_BMOb(x) –b

σ,

H≤

∞

|x|+j

|x–y|<t m

i=

bi(x) –bi(y)

(x–y,x)

|x–y|n–– –

(x,x)

|x|n–– aj(y)dy



dt t

/

≤

Bj



|x|+ j

_|_x(_–x_y–_|ny––,x) –

(x,x)

|x|n––

m

i=

bi(x) –bi(y)aj(y)dy

≤C  |x|+ j

m

i=

σ∈C_im

b(x) –b_σ

Bj

(x–y,x)

|x–y|n–– –

(x,x)

|x|n––

aj(y)b(y) –b

σcdy

≤C  |x|+ j

m

i=

σ∈Cm i

b(x) –b_σ

Bj

(x–y,x)

|x–y|n–– –

(x,x)

|x|n–– rdy

/r

×

Bj

aj(y)b(y) –b

σc t

dy

/t

≤C  |x|+ j

m

i=

σ∈C_im

aLq|Bj|/t–/qbσcBMOb(x) –b

σ

 j+

Bj

ωr(δ)

δ dδ

≤C

jn(–/q+/t–α)

|x|+ j m

i=

σ∈Cm_i

bσc_BMOb(x) –b

σ,

thus

μb(aj)χk_Lq_

≤Cj(/++n(–/q–α))

m

i=

σ∈Cm i

bσc_BMO

Ck

|x|–(n+/)b(x) –b_σq /q

+Cjn(–/q–α) m

i=

σ∈C_im

bσcBMO

Ck

|x|–b(x) –b_σq

/q

≤Cj(/++n(–/q)–α)+kn(/q–/n–)_b BMO

+Cjn(–/q–α)+kn(/q–/n–/n)_b BMO.

For the sake of simplicity, we denote

W(j,k)

= j(/++n(–/q)–α)+kn(/q–/n–)

+jn(–/q–α)+kn(/q–/n–/n)_,

then

(9)

we obtain

J≤Cbp_BMO

∞

k=–∞ kαp

_k_–

j=–∞

|λj|W(j,k) p

≤ ⎧ ⎨ ⎩

Cbp_BMO∞_j_=–_∞|λj|p

∞

k=j+W(j,k)p,  <p≤,

Cbp_BMO_j∞_=–_∞|λj|p[

∞

k=j+W(j,k)p/][

∞

k=j+W(j,k)p _/

]p/p_, _p_{> }

≤Cbp_BMO

∞

j=–∞

|λj|p≤CbpBMOf p HK˙α,p

q,b

.

This completes the proof of Theorem .

Remark Theorem  also holds for nonhomogeneous Herz-type spaces.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

In this paper, WY carried out the (H_bp,Lp_{) and (}_H_K˙α,p q,b,K˙

α,p

q )-type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels. YH, QL, JZ, XW participated in the analysis. All authors read and revised the ﬁnal manuscript.

Author details

1_{College of Electrical and Information Engineering, Hunan University, Changsha, Hunan 410082, P.R. China.}2_{School of}

Electrical and Automation Engineering, Hefei University of Technology, Hefei, Anhui 230009, P.R. China.

Acknowledgements

This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant No. 50925727, National Natural Science Foundation of China under Grant No. 60876022, The National Defense Advanced Research Project Grant No. C1120110004, Hunan Provincial Science and Technology Foundation of China under Grant No. 2010J4 and 2011JK2023, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018.

Received: 30 March 2012 Accepted: 3 December 2012 Published: 27 December 2012 References

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doi:10.1186/1029-242X-2012-308