Multilinear fractional integral operators on central Morrey spaces with variable exponent

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R E S E A R C H

Open Access

Multilinear fractional integral operators

on central Morrey spaces

with variable exponent

Hongbin Wang

1*

_{and Jingshi Xu}

2

*_{Correspondence:}

[email protected] 1_{School of Mathematics and}

Statistics, Shandong University of Technology, Zibo, China Full list of author information is available at the end of the article

Abstract

In this paper, we obtain the boundedness of the multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent.

MSC: 42B35; 42B20; 46E30

Keywords: Multilinear fractional integral operator; Commutator; Variable exponent;

λ

-centralBMOspaces; Central Morrey space

1 Introduction

The study of multilinear integral operators was motivated not only as the generalization of the theory of linear ones but also their natural appearance in analysis. It has increas-ing attention and much development in recent years, such as the study of the bilinear Hilbert transform by Lacey and Thiele [21,22] and the systematic treatment of multilinear Calderón–Zygmund operators by Grafokas and Torres [13,14], Grafokas and Kalton [12]. The importance of fractional integral operators is owing to the fact that they are smooth operators and have been extensively used in various areas such as potential analysis, har-monic analysis and partial diﬀerential equations. As one of the most important operators, the multilinear fractional integral operator (also known as the multilinear Riesz potential) has also attracted more attention, see for example [3,11,18,24].

It is well known that function spaces with variable exponent arouse strong interest not only in harmonic analysis but also in applied mathematics. The theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník [19] in 1991. Lebesgue and Sobolev spaces with integrabil-ity exponent have been widely studied, see [5,8] and the references therein. Many applica-tions of these spaces were given, for example, in the modeling of electrorheological ﬂuids [26], in the study of image processing [4], and in diﬀerential equations with nonstandard growth [15]. On the other hand, theλ-central bounded mean oscillation spaces, Morrey type spaces and related function spaces have interesting applications in studying bound-edness of operators including singular integral operators; see for example [1,9,19,27–29]. In 2015, Mizuta, Ohno and Shimomura introduced the non-homogeneous central

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rey spaces of variable exponent in [23]. Recently, Fu et al. introduced theλ-central BMO spaces and the central Morrey spaces with variable exponent and gave the boundedness of some operators in [10]. In [2,6,7,17] and [31–34], the authors proved the boundedness of some integral operators on variable function spaces, respectively. Meanwhile, some au-thors gave the boundedness of multilinear integral operators and their commutators on variable exponent function spaces, such as [16,30,35].

Motivated by [9,10,29], we will study the boundedness of the multilinear fractional integral operators and their commutators on the central Morrey spaces with variable ex-ponent.

Let us explain the outline of this article. In Sect.2, we first briefly recall some standard notations and lemmas in variable Lebesgue spaces. Then we will recall the definiton of the

λ-central BMO spaces and central Morrey spaces with variable exponent. In Sect.3, we will establish the boundedness for a class of multi-sublinear fractional integral operators on central Morrey spaces with variable exponent. Subsequently the boundedness of mul-tilinear fractional integral commutators on central Morrey spaces with variable exponent will be obtained in Sect.4. In Sect.5, we will also consider the boundedness of another multilinear fractional integral commutators.

In addition, we denote the Lebesgue measure and the characteristic function of a mea-surable setA⊂Rn_by_|_A_|_and_χ

A, respectively. The notationf ≈gmeans that there exist constantsC1,C2> 0 such thatC1g≤f ≤C2g.

2 Variable exponent function spaces

Firstly we give some notation and basic deﬁnitions on variable exponent Lebesgue spaces. Given an open setE⊂Rn_{, and a measurable function}_p₍_·_{) :}_E_→_[1,_∞_)._p₍_·_{) is the} con-jugate exponent deﬁned byp(·) =p(·)/(p(·) – 1).

The setP(E) consists of allp(·) :E→[1,∞) satisfying

p–=ess infp(x) :x∈E> 1, p+=ess supp(x) :x∈E<∞.

ByLp(·)₍_E_{) we denote the space of all measurable functions}_f _on_E_{such that, for some}

λ> 0,

E

_|

f(x)|

λ p(x)

dx<∞.

This is a Banach function space with respect to the Luxemburg–Nakano norm,

f _Lp(·)₍_E₎=inf

λ> 0 :

E

_|

f(x)|

λ p(x)

dx≤1

.

The space Lp_loc(·)(Ω) is deﬁned by Lp_loc(·)(Ω) :={f :f ∈Lp(·)(E) for all compact subsets E⊂Ω}.

Letf ∈L1

loc(Rn), the Hardy–Littlewood maximal operator is deﬁned by

Mf(x) =sup r>0

1

|Br(x)|

Br(x)

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whereBr(x) ={y∈Rn:|x–y|<r}. The setB(Rn) consists ofp(·)∈P(Rn) satisfying the condition thatMis bounded onLp(·)(Rn).

In variableLpspaces there are some important lemmas as follows.

Lemma 2.1([7]) If p(·)∈P(Rn₎_{and satisﬁes}

p(x) –p(y) ≤ C

–log(|x–y|), |x–y| ≤1/2, (2.1)

and

p(x) –p(y) ≤ C

log(|x|+e), |y| ≥ |x|, (2.2)

then p(·)∈B(Rn_), _{that is}_, _{the Hardy–Littlewood maximal operator M is bounded on}

Lp(·)(Rn).

Lemma 2.2([20] (Generalized Hölder inequality)) Let p(·)∈P(Rn_)._{If f} _∈_Lp(·)₍_Rn₎_and

g∈Lp(·)_(Rn_),_{then fg is integrable on}_Rn_and

Rn

f(x)g(x) dx≤rp f Lp(·)₍_Rn₎ g _Lp(·)₍_Rn₎,

where

rp= 1 + 1/p–– 1/p+.

Lemma 2.3([17]) Suppose p(·)∈B(Rn_)._{Then there exists a positive constant C such that}_,

for all balls B inRn_,

1

|B| χB Lp(·)(Rn) χB Lp(·)₍_Rn₎≤C.

Lemma 2.4([17]) Let p(·)∈B(Rn_)._{Then there exists a positive constant C such that}_,_for

all balls B inRn_{and all measurable subsets S}_⊂_B_,

χB Lp(·)(Rn₎ χS Lp(·)₍_Rn₎ ≤

C|B|

|S|,

χS Lp(·)(Rn₎ χB Lp(·)₍_Rn₎ ≤

C

_|

S|

|B|

δ1

and

χS _Lp(·)₍_Rn₎ χB _Lp(·)₍_Rn₎

≤C

|S|

|B|

δ2

,

whereδ1,δ2are constants with0 <δ1,δ2< 1.

Lemma 2.5([8]) Let p(·)∈P(Rn₎_{satisﬁes conditions}₍_2.1₎_and₍_2.2₎_{in Lemma}_2.1_._Then

χQ Lp(·)(Rn₎≈ ⎧ ⎨ ⎩

|Q|p(1x) _if|_Q| ≤₂n_{and x}∈_Q_,

|Q|p(1∞) _if|_Q| ≥₁

for every cube(or ball)Q⊂Rn_,_{where p}₍_∞_{) =}_lim

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Lemma 2.6([8]) Let p(·),q(·),s(·)∈P(Rn₎_{be such that}

1 s(x)=

1 p(x)+

1 q(x)

for almost every x∈Rn_._Then

fg _Ls(·)₍_Rn₎≤2 f _Lp(·)₍_Rn₎ g _Lq(·)₍_Rn₎

for all f ∈Lp(·)₍_Rn₎_{and g}_∈_Lq(·)₍_Rn_).

Now we recall that the central Morrey space with variable exponent and theλ-central bounded mean oscillation space with variable exponent in [10] are deﬁned as follows.

Deﬁnition 2.1([10]) Letq(·)∈P(Rn_{) and}_λ_∈_{R. The central Morrey space with variable} exponentB˙q(·),λ_(Rn_{) is deﬁned by}

˙

Bq(·),λ_Rn₌_f _∈_Lq(·) loc

Rn_: _f

˙

Bq(·),λ₍_Rn₎<∞

,

where

f _B˙q(·),λ₍_Rn₎=sup

R>0

fχB(0,R) Lq(·)₍_Rn₎ |B(0,R)|λ _χ

B(0,R) Lq(·)₍_Rn₎

.

Deﬁnition 2.2([10]) Letq(·)∈P(Rn_{) and}_λ_{< 1/}_n_{. The}_λ_{-central BMO space with} vari-able exponent CBMOq(·),λ₍_Rn_{) is deﬁned by}

CBMOq(·),λ_Rn₌_f _∈_Lq(·) loc

Rn_: _f

CBMOq(·),λ(Rn₎<∞

,

where

f _CBMOq(·),λ₍_Rn₎=sup

R>0

(f –fB(0,R))χB(0,R) Lq(·)₍_Rn₎ |B(0,R)|λ _χ

B(0,R) Lq(·)₍_Rn₎

.

Remark2.1 Denote byBq(·),λ_(Rn_{) and CMO}q(·),λ_(Rn_{) the inhomogeneous versions of the} central Morrey space and theλ-central BMO space with variable exponent, which are de-fined, respectively, by taking the supremum overR≥1 in Definition2.1and Definition2.2

instead ofR> 0 there.

Remark2.2 Our results in this paper remain true for the inhomogeneous versions ofλ -central BMO spaces and -central Morrey spaces with variable exponent.

3 Multilinear fractional integral operators

Letm∈NandK(y0,y1, . . . ,ym) be a function deﬁned away from the diagonaly0=y1=

· · ·=ymin (Rn)m+1. We denote byfthem-tuple (f1, . . . ,fm). Now we consider thatT is an

m-linear operator deﬁned on the product of test functions such that, forK, the integral representation below is valid:

T(f)(x) =

Rn· · ·

RnK(x,y1, . . . ,ym)

m

j=1

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whenever fj, j = 1, . . . ,m, are smooth functions with compact support and x ∈/

m j=1suppfj.

Particularly, there is a kind of multilinear operatorTα,m, which is called multilinear frac-tional integral operator, whose kernel is

K(x,y1, . . . ,ym) = (x–y1, . . . ,x–ym)

α–mn

, 0 <α<mn. (3.1)

In 1999, Kenig and Stein [18] gave the boundedness of the above multilinear fractional integral operatorTα,mon the product of Lebesgue spaces.

Theorem A([18]) Let0 <α<mn and Tα,m be an m-linear fractional integral operator

with kernel K satisfying(3.1).Suppose1≤p1,p2, . . . ,pm≤ ∞, 1/q= 1/p1+· · ·+ 1/pm–

α/n> 0.

(1) If eachpj> 1,j= 1, . . . ,m,then

Tα,m(f)_Lq₍_Rn₎≤C

m

i=1

fi Lpi(Rn₎.

(2) If eachpj= 1for somej,then

Tα,m(f)_Lq,∞₍_Rn₎≤C

m

i=1

fi Lpi(Rn₎.

In the variable exponent case, Tan, Liu and Zhao [30] gave the following result.

Theorem B([30]) Let m∈N, 0 <α<mn,q(·),p1(·), . . . ,pm(·)∈P(Rn)satisfy conditions (2.1)and(2.2)in Lemma2.1and1/q(·) = 1/p1(·) +· · ·+ 1/pm(·) –α/n.Then

Tα,m(f)_Lq(·)₍_Rn₎≤C

m

i=1

fi Lpi(·)₍_Rn₎.

Next we will give the boundedness of a class of multi-sublinear fractional integral oper-atorsT on the product of central Morrey spaces with variable exponent.

Theorem 3.1 Let m∈N, 0 <α<mn and T be a multi-sublinear fractional integral oper-ator such that

T(f)(x) ≤C

(Rn₎m

|f1(y1)| · · · |fm(ym)|

|(x–y1, . . . ,x–ym)|mn–α

dy1· · ·dym (3.2)

for any integrable functions f1, . . . ,fmwith compact support and x∈/

m

j=1suppfj.Suppose

λj< –_mnα , λ=

m

j=1λj+α/n, pj(·) (j= 1, . . . ,m),q(·)∈P(Rn)satisfy conditions(2.1)and (2.2)in Lemma2.1and1/q(·) =m_j₌₁1/pj(·) –α/n> 0.If T is bounded from Lp1(·)(Rn)×

· · ·×Lpm(·)_(Rn₎_{into L}q(·)_(Rn_),_{then T is also bounded from}_B˙p1(·),λ1(Rn₎×· · ·× ˙Bpm(·),λm_(Rn₎

intoB˙q(·),λ_(Rn_).

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Corollary 3.1 Let m∈N, 0 <α<mn and Tα,mbe an m-linear fractional integral operator

with kernel K satisfying(3.1).Supposeλj< –_mnα ,λ=

m

j=1λj+α/n,pj(·) (j= 1, . . . ,m),q(·)∈

P(Rn₎_{satisfy conditions}₍_2.1₎_and₍_2.2₎_{in Lemma}_2.1_and_1/_q₍_·_{) =}m

j=11/pj(·) –α/n> 0.

Then Tα,mis bounded fromB˙p1(·),λ1(Rn)× · · · × ˙Bpm(·),λm(Rn)intoB˙q(·),λ(Rn).

Proof of Theorem3.1 In order to simplify the proof, we consider only the situation when m = 2. Actually, a similar procedure works for all m∈N. Let f1, f2 be functions in

˙

Bp1(·),λ1₍Rn_{) and}B˙p2(·),λ2₍Rn_{), respectively. For ﬁxed}_R_{> 0, denote}_B_(0,_R_{) by}_B_{. We need}

to prove

T(f1,f2)χB_Lq(·)₍_Rn₎≤C|B| λ

χB Lq(·)₍_Rn₎ f1 _B˙p₁₍·),λ₁₍_Rn₎ f2 _B˙p₂₍·),λ₂₍_Rn₎,

whereCis a constant independent ofR. By the Minkowski inequality we write

T(f1,f2)χB_Lq(·)₍_Rn₎≤T(f1χ2B,f2χ2B)χB_Lq(·)₍_Rn₎

+T(f1χ(2B)c,f₂χ₂_B)χ_B

Lq(·)₍_Rn₎

+T(f1χ2B,f2χ(2B)c)χ_B

Lq(·)₍_Rn₎

+T(f1χ(2B)c,f₂χ₍₂_B₎c)χ_B

Lq(·)₍_Rn₎

=:I1+I2+I3+I4. (3.3)

We ﬁrst estimate I1. Using Lemma2.4and the boundedness of T from Lp1(·)(Rn)×

Lp2(·)(Rn_{) into}_Lq(·)(Rn_{), we have}

I1≤C f1χ2B Lp1(·)₍_Rn₎ f2χ2B Lp2(·)₍_Rn₎

≤C f1 _B˙p1(·),λ1(Rn₎|2B|λ1 χ2B Lp1(·)₍_Rn₎ f2 _B˙p2(·),λ2(Rn₎|2B|λ2 χ2B Lp2(·)₍_Rn₎ ≤C|2B|λ1+λ2+α/n|₂_B|–α/n _f

1 _B˙p1(·),λ1₍_Rn₎ f2 _B˙p2(·),λ2₍_Rn₎ χ2B _Lp1(·)₍_Rn₎ χ2B _Lp2(·)₍_Rn₎ ≤C|2B|λ1+λ2+α/n _f

1 _B˙p₁₍·),λ₁₍_Rn₎ f2 _B˙p₂₍·),λ₂₍_Rn₎ χ2B Lq(·)₍_Rn₎ ≤C|B|λ _χ

2B Lq(·)(Rn₎ f1 _B˙p₁₍·),λ1(Rn₎ f2 _B˙p₂₍·),λ2(Rn₎, (3.4)

where

χ2B _Lp₁₍·)₍_Rn₎ χ2B _Lp₂₍·)₍_Rn₎≈ |2B|1/p1(·)+1/p2(·)=|2B|1/q(·)+α/n≈ |2B|α/n χ2B Lq(·)₍_Rn₎.

Next we estimate I2. Noting that |(x–y1,x–y2)|2n–α ≥ |x–y1|2n–α, by using (3.2),

λ1< –₂α_n, Lemma2.3, the Minkowski inequality and the generalized Hölder inequality,

we have

I2=T(f1χ(2B)c,f₂χ₂_B)χ_B

Lq(·)₍_Rn₎ ≤

∞

k=1

T(f1χ2k+1_B_\₂k_B,f₂χ₂_B)χ_B

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≤C

∞

k=1

₂_B₂k+1_B_\₂k_B

|f1(y1)||f2(y2)|

|(·–y1,·–y2)|2n–α

dy1dy2χB(·)

Lq(·)₍_Rn₎ ≤C f2χ2B L1₍_Rn₎

∞

k=1

2k+1_B_\₂k_B

|f1(y1)|

| ·–y1|2n–α

dy1χB(·)

Lq(·)₍_Rn₎ ≤C f2χ2B Lp2(·)₍_Rn₎ χ2B

Lp2(·)₍_Rn₎ ∞

k=1

χB Lq(·)₍_Rn₎

2k–1R–2n+α f1χ2k+1_B _L1₍_Rn₎ ≤C f2χ2B Lp2(·)₍_Rn₎ χ2B

Lp2(·)₍_Rn₎ χB Lq(·)₍_Rn₎ ×

∞

k=1

2kB –2+α/n f1χ2k+1_B _Lp1(·)₍_Rn₎ χ₂k+1_B

Lp1(·)(Rn₎ ≤C f2 _B˙p2(·),λ2(Rn₎|2B|λ2 χ2B Lp2(·)₍_Rn₎ χ2B _Lp₂₍·)

(Rn₎ χB Lq(·)₍_Rn₎|B|–2+α/n ×

∞

k=1

2kn(–2+α/n) f1 _B˙p1(·),λ1(Rn₎ 2k+1B λ1

χ₂k+1_B _Lp1(·)₍_Rn₎ χ₂k+1_B

Lp1(·)(Rn₎ ≤C f1 _B˙p₁₍·),λ1(Rn₎ f2 _B˙p₂₍·),λ2(Rn₎|B|λ2+1–2+α/n+λ1+1 χB Lq(·)(Rn₎

× ∞

k=1

2–2kn+kα+nλ2+n+(k+1)nλ1+(k+1)n

≤C f1 _B˙p₁₍·),λ₁₍_Rn₎ f2 _B˙p₂₍·),λ₂₍_Rn₎|B|λ χB Lq(·)₍_Rn₎ ∞

k=1

2kn(–1+λ1+α/n)

≤C f1 _B˙p1(·),λ1₍_Rn₎ f2 _B˙p2(·),λ2₍_Rn₎|B|λ χB Lq(·)₍_Rn₎ ∞

k=1

2kn(λ1+2αn)

≤C|B|λ χB Lq(·)₍_Rn₎ f1 _B˙p1(·),λ1₍_Rn₎ f2 _B˙p2(·),λ2₍_Rn₎. (3.5)

Similarly, we estimate I3. Noticing that |(x–y1,x–y2)|2n–α ≥ |x–y2|2n–α, by (3.2),

λ2< –₂α_n, Lemma2.3, the Minkowski inequality and the generalized Hölder inequality,

we obtain

I3=T(f1χ2B,f2χ(2B)c)χ_B

Lq(·)₍_Rn₎ ≤

∞

k=1

T(f1χ2B,f2χ2k+1_B_\₂k_B)χB_Lq(·)₍_Rn₎

≤C

∞

k=1

₂k+1_B_\₂k_B

2B

|f1(y1)||f2(y2)|

|(·–y1,·–y2)|2n–α

dy1dy2χB(·)

Lq(·)₍_Rn₎ ≤C f1χ2B L1₍_Rn₎

∞

k=1

₂k+1_B_\₂k_B

|f2(y2)|

| ·–y2|2n–α

dy2χB(·)

Lq(·)₍_Rn₎ ≤C f1χ2B Lp1(·)₍_Rn₎ χ2B

Lp1(·)₍_Rn₎ ∞

k=1

χB Lq(·)₍_Rn₎

2k–1R–2n+α f2χ2k+1_B _L1₍_Rn₎ ≤C f1χ2B Lp1(·)₍_Rn₎ χ2B

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× ∞

k=1

Lp2(·)₍_Rn₎ ≤C f1 _B˙p1(·),λ1(Rn₎|2B|λ1 χ2B Lp1(·)₍_Rn₎ χ2B

Lp1(·)₍_Rn₎ χB Lq(·)₍_Rn₎|B|–2+α/n ×

∞

k=1

2kn(–2+α/n) f2 _B˙p2(·),λ2(Rn₎ 2k+1B λ2

χ₂k+1_B _Lp2(·)₍_Rn₎ χ₂k+1_B

Lp2(·)₍_Rn₎ ≤C f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎|B|λ1+1–2+α/n+λ2+1 χB Lq(·)₍_Rn₎

× ∞

k=1

2–2kn+kα+nλ1+n+(k+1)nλ2+(k+1)n

≤C f1 _B˙p₁₍·),λ1(Rn₎ f2 _B˙p₂₍·),λ2(Rn₎|B|λ χB Lq(·)₍_Rn₎ ∞

k=1

2kn(–1+λ2+α/n)

≤C|B|λ _χ

B Lq(·)₍_Rn₎ f₁ _B_˙p₁₍·),λ₁₍_Rn₎ f2 _B˙p₂₍·),λ₂₍_Rn₎. (3.6)

For the estimate ofI4. Noting that|(x–y1,x–y2)|2n–α≥ |x–y1|n–α/2|x–y2|n–α/2, by

(3.2),λj< –₂α_n,j= 1, 2, Lemma2.3, the Minkowski inequality and the generalized Hölder inequality, we have

I4=T(f1χ(2B)c,f₂χ₍₂_B₎c)χ_B

Lq(·)₍_Rn₎ ≤

∞

k1=1

∞

k2=1

_T₍_f₁χ₂k₁₊₁_B_\₂k₁_B,f₂χ₂k₂₊₁_B_\₂k₂_B)χ_B

Lq(·)₍_Rn₎

≤C ∞ k=1 ∞

k2=1

₂k2+1B\2k2B

2k1+1B\2k1B

|f1(y1)||f2(y2)|

|(·–y1,·–y2)|2n–α

dy1dy2χB(·)

Lq(·)₍_Rn₎ ≤C ∞ k=1 ∞

k2=1

2 j=1

2kj+1B\2kjB

|fj(yj)|

| ·–yj|n–α/2

dyjχB(·)

Lq(·)(Rn₎ ≤C 2 j=1 _∞

kj=1

2kj–1_R–n+α/2

2kj+1B

fj(yj) dyj

χB Lq(·)₍_Rn₎

≤C χB Lq(·)₍_Rn₎

2

j=1

_∞

kj=1

2kj_B –1+2αn _f

jχ₂kj+1

B Lpj(·)(Rn₎ χ₂kj+1_B

Lp

j(·)₍_Rn₎

2

j=1

fj _B˙pj(·),λj₍_Rn₎ ∞

kj=1

2kj_B λj–1+

α

2n _χ

2kj+1B Lpj(·)(Rn₎ χ₂kj+1_B

Lp

j(·)₍_Rn₎

2

j=1

fj _B˙pj(·),λ_j

(Rn₎ ∞

kj=1

2kj_B λj–1+2αn ₂kj_B

2

j=1

fj _B_˙pj(·),λj₍_Rn₎|B| λ

∞

kj=1

2kjn(λj+2αn)

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Combining the estimates of (3.3)–(3.7), we have

T(f1,f2)χB_Lq(·)₍_Rn₎≤C|B| λ _χ

B Lq(·)₍_Rn₎ f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎,

that is,

_T₍_f₁_,_f₂₎_˙

Bq(·),λ₍_Rn₎≤C f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎.

This completes the proof of Theorem3.1.

4 Multilinear fractional integral commutators

Letm∈N,b= (b1,b2, . . . ,bm) andbi∈CBMOui(·),vi(Rn),i= 1, . . . ,m. Then the multilinear commutators of fractional integral operator are deﬁned by

[b,Tα,m](f)(x) =

(Rn₎m m

i=1(bi(x) –bi(yi)fi(yi)

|(x–y1, . . . ,x–ym)|mn–α

dy1· · ·dym. (4.1)

Theorem 4.1 Let0 <α<mn, 0 <vi< 1/n,λi< –_mnα ,λ=

m i=1vi+

m

i=1λi+α/n,vi+λi< –α/n, pi(·) (i= 1, . . . ,m), q(·)∈P(Rn)satisfy conditions (2.1)and (2.2)in Lemma 2.1 1/p(·) =m_i₌₁1/pi(·) –α/n> 0and1/q(·) =

m

i=11/ui(·) +

m

i=11/pi(·) –α/n.Then[b,Tα,m]

is also bounded fromB˙p1(·),λ1(Rn₎× · · · × ˙Bpm(·),λm_(Rn₎_into_B˙q(·),λ_(Rn₎_{and the following}

inequality holds:

[b,Tα,m]f_B˙q(·),λ₍_Rn₎≤C

m

i=1

bi _CBMOui(·),_vi₍_Rn₎ fi _B˙pi(·),λi₍_Rn₎

.

Proof Without loss of generality, we still consider only the situation whenm= 2. Letf1,f2

be functions inB˙p1(·),λ1(Rn_{) and}_B˙p2(·),λ2(Rn_{), respectively. For ﬁxed}_R_{> 0, denote}_B_(0,_R₎ byB. We have the following decomposition:

[b,Tα,2]f(x) =

b1–{b1}B

b2–{b2}B

Tα,2(f1,f2)(x)

–b1–{b1}B

Tα,2

f1,

b2(·) –{b2}B

f2

(x)

–b2–{b2}B

Tα,2

b1(·) –{b1}B

f1,f2

(x)

+Tα,2

b1(·) –{b1}B

f1,

b2(·) –{b2}B

f2

(x)

=:J1+J2+J3+J4. (4.2)

Thus, by the Minkowski inequality we write

[b,Tα,2]fχB_Lq(·)₍_Rn₎≤

4

i=1

JiχB Lq(·)₍_Rn₎=:

4

i=1

Li. (4.3)

Because of the symmetry off1 andf2, we can see that the estimate ofL2is analogous to

that ofL3. Then we will estimateL1,L2andL4, respectively.

(10)

(i) ForL1, by using the Minkowski inequality we can write

L1≤b1–{b1}B

b2–{b2}B

Tα,2(f1χ2B,f2χ2B)(·)χB(·)_Lq(·)₍_Rn₎

+b1–{b1}B

b2–{b2}B

Tα,2(f1χ2B,f2χ(2B)c)(·)χ_B(·)

Lq(·)₍_Rn₎

+b1–{b1}B

b2–{b2}B

Tα,2(f1χ(2B)c,f₂χ₂_B)(·)χ_B(·)

Lq(·)₍_Rn₎

+b1–{b1}B

b2–{b2}B

Tα,2(f1χ(2B)c,f₂χ₍₂_B₎c)(·)χ_B(·)

Lq(·)₍_Rn₎

=:L11+L12+L13+L14. (4.4)

Firstly we estimateL11. Noticing that _p1₍_·₎ =

2

i=1pi1(·)– α

n, then

1

q(·) =

2

i=1ui1(·) +

1

p(·). By

Lemma2.6and using the boundedness ofTα,2fromLp1(·)(Rn)×Lp2(·)(Rn) intoLq(·)(Rn) in

TheoremB, we get

L11≤CTα,2(f1χ2B,f2χ2B)(·)χB(·)_Lp(·)₍_Rn₎

2

i=1

bi–{bi}B

χB_Lui(·)₍_Rn₎

≤C

2

i=1

fiχB _Lpi(·)₍_Rn₎

2

i=1

bi–{bi}B

χB_Lui(·)₍_Rn₎ ≤C|B|λ1+λ2+v1+v2 _χ

B Lp1(·)(Rn₎ χB Lp2(·)(Rn₎ χB Lu1(·)(Rn₎ χB Lu2(·)(Rn₎ ×

2

i=1

fi _B˙pi(·),λ_i₍_Rn₎ bi _CBMOui(·),_vi₍_Rn₎

≤C|B|λ1+λ2+v1+v2+α/n+1/q(·) 2

i=1

fi _B˙pi(·),λi(Rn₎ bi CBMOui(·),vi(Rn₎

≤C|B|λ+1/q(·)

2

i=1

fi _B˙pi(·),λi₍_Rn₎ bi _CBMOui(·),_vi₍_Rn₎

, (4.5)

where

χB Lp1(·)(Rn₎ χB Lp2(·)(Rn₎ χB Lu1(·)(Rn₎ χB Lu2(·)(Rn₎≈ |B|1/p1(·)+1/p2(·)+1/u1(·)+1/u2(·)

=|B|1/q(·)+α/n.

Now we estimate L12. Noticing that|(x–y1,x–y2)|2n–α≥ |x–y2|2n–α. Byλ2 < –₂α_n,

Lemma2.3and the generalized Hölder inequality, we obtain

Tα,2(f1χ2B,f2χ(2B)c)(x)

=

(Rn₎2

[f1(y1)χ2B(y1)][f2(y2)χ(2B)c(y₂)] |(x–y1,x–y2)|2n–α

dy1dy2

≤C

2B

_f₁(y1) dy1

(2B)c

|f2(y2)|

|x–y2|2n–α

dy2

≤C f1χ2B Lp1(·)₍_Rn₎ χ2B

Lp1(·)₍_Rn₎ ∞

k=1

2k+1_B_\₂k_B

|f2(y2)|

|x–y2|2n–α

dy2

≤C f1 _B˙p1(·),λ1(Rn₎|2B|λ1 χ2B Lp1(·)₍_Rn₎ χ2B

(11)

× ∞

k=1

2k–1R–2n+α

2k+1_B

f2(y2) dy2

≤C f1 _B˙p1(·),λ1(Rn₎|2B|λ1+1 ∞

k=1

Lp2(·)₍_Rn₎ ≤C f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎|B|λ1+1–2+α/n+1+λ2

∞

k=1

2kn(–2+α/n+λ2+1)

≤C f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎|B|λ1+λ2+α/n ∞

k=1

2kn(2αn+λ2)

≤C|B|λ1+λ2+α/n

2

i=1

fi _B˙pi(·),λi(Rn₎.

Let _u1₍_·₎ =_u1

1(·)+

1

u2(·), by the fact 1/p(·) =

2

i=11/pi(·) –α/n> 0 and 1/q(·) =

m

i=11/ui(·) +

m

i=11/pi(·) –α/n, thenu(·) >q(·). Thus by Lemma2.5and Lemma2.6we get

b1–{b1}B

b2–{b2}B

χB_Lq(·)₍_Rn₎ ≤Cb1–{b1}B

χB_Lu₁₍·)₍_Rn₎b2–{b2}B

χB_Lu₂₍·)₍_Rn₎ χB Lp(·)₍_Rn₎ ≤C b1 _CBMOu1(·),v1₍_Rn₎|B|v1 χB _Lu1(·)₍_Rn₎

× b2 CBMOu2(·),v2(Rn₎|B|v2 χB Lu2(·)(Rn₎ χB Lp(·)₍_Rn₎

≤C|B|v1+v2+

1

u1(·)+ 1

u2(·)+ 1

p(·)

2

i=1

bi _CBMOui(·),_vi₍_Rn₎

≤C|B|v1+v2+_q1₍_·₎

2

i=1

bi _CBMOui(·),_vi₍_Rn₎.

This yields

L12≤C|B|

λ1+λ2+α/n+v1+v2+q1(·)

2

i=1

≤C|B|λ+q1(·)

2

i=1

bi _CBMOui(·),_vi₍_Rn₎ fi _B˙pi(·),λ_i₍_Rn₎

. (4.6)

Similarly, we have

L13≤C|B|

λ+_q1₍_·₎2

i=1

bi _CBMOui(·),_vi₍_Rn₎ fi _B˙pi(·),λ_i₍_Rn₎

. (4.7)

Now for the estimate ofL14. Note that|(x–y1,x–y2)|2n–α≥ |x–y1|n–α/2|x–y2|n–α/2.

Usingλj< –₂α_n,j= 1, 2 and the generalized Hölder inequality, we have

Tα,2(f1χ(2B)c,f₂χ₍₂_B₎c)(x)

≤C

Rn

|f1(y1)χ(2B)c(y₁)||f₂(y₂)_χ₍₂_B₎c(y₂)| |(x–y1,x–y2)|2n–α

(12)

≤C

∞

k1=1

∞

k2=1

2k2+1B\2k2B

2k1+1B\2k1B

|f1(y1)||f2(y2)|

|(x–y1,x–y2)|2n–α

dy1dy2

≤C

∞

k1=1

∞

k2=1

2

i=1

2ki+1_B_\₂_ki_B

|fi(yi)|

|(x–yi)|n–α/2

dyi ≤C 2 i=1 ∞

ki=1

2ki–1_R–n+α/2

2ki+1B

fi(yi) dyi

≤C 2 i=1 ∞

ki=1

2kj_B –1+2αn _f

jχ₂kj+1

B Lpj(·)(Rn₎ χ₂kj+1_B

Lpj(·)₍_Rn₎ ≤C

2

i=1

fj _B_˙pj(·),λ_j

(Rn₎ ∞

ki=1

2kj_B –1+2αn+λj+1

≤C|B|λ1+λ2+α/n

2

i=1

fi _B˙pi(·),λi₍_Rn₎.

Similar to the estimates forL12, we have

L14≤C|B|

λ+_q1₍_·₎ 2

i=1

bi CBMOui(·),vi(Rn₎ fi _B˙pi(·),λi(Rn₎

. (4.8)

By the estimates ofL1j,j= 1, 2, 3, 4, we get

L1≤C|B|

λ+_q1₍_·₎ 2

i=1

bi CBMOui(·),_vi₍_Rn₎ fi _B˙pi(·),λi(Rn₎

. (4.9)

(ii) ForL2, we obtain

L2≤b1–{b1}B

Tα,2

f1χ2B,

b2(·) –{b2}B

f2χ2B

χB_Lq(·)₍_Rn₎

+b1–{b1}B

Tα,2

f1χ(2B)c,b₂–{b₂}_Bf₂χ₂_Bχ_B

Lq(·)(Rn₎

+b1–{b1}B

Tα,2

f1χ2B,

b2–{b2}B

f2χ(2B)cχ_B

Lq(·)₍_Rn₎

+b1–{b1}B

Tα,2

f1χ(2B)c,b₂–{b₂}_Bf₂χ₍₂_B₎cχ_B(·)

Lq(·)₍_Rn₎

=:L21+L22+L23+L24. (4.10)

Let _q1₍_·₎=_u1

1(·)+

1

q1(·),

1

q1(·)=

1

p1(·)+

1

g(·)–

α

n,

1

g(·) = 1

p2(·)+

1

u2(·). Byλj< –

α

2n,j= 1, 2 and bound-edness ofTα,2fromLp1(·)(Rn)×Lg(·)(Rn) intoLq1(·)(Rn), we get

L21≤Cb1–{b1}B

χB_Lu1(·)₍_Rn₎Tα,2

f1χ2B,

b2(·) –{b2}B

f2χ2B

χB_Lq1(·)₍_Rn₎ ≤Cb1–{b1}B

χB_Lu₁₍·)₍_Rn₎ f1χ2B Lp1(·)(Rn₎b2–{b2}B

f2χ2B_Lg(·)₍_Rn₎ ≤Cb1–{b1}B

χB_Lu₁₍·)₍_Rn₎ f1χ2B _Lp1(·)₍_Rn₎ f2χ2B _Lp2(·)₍_Rn₎ ×b2–{b2}2B+{b2}2B–{b2}B

χ2B_Lu2(·)₍_Rn₎

(13)

× f2 _B˙p2(·),λ2(Rn₎|2B|λ2 χ2B Lp2(·)₍_Rn₎ b2 CBMOu2(·),v2(Rn₎|2B|v2 χ2B Lu2(·)₍_Rn₎

≤C|B|v1+v2+λ1+λ2+u₁₍1·)+u₂₍1·)+p₁₍1·)+p₂₍1·)

2

i=1

bi CBMOui(·),vi(Rn₎ fi _B˙pi(·),λi(Rn₎

≤C|B|λ+1/q(·)

2

i=1

, (4.11)

where

_{b2}2B–{b2}B ≤ 1

|B|b2–{b2}2B

χ2B_Lu2(·)₍_Rn₎ χ2B _Lu₂₍·)

(Rn₎ ≤Cb2–{b2}2B

χ2B_Lu2(·)₍_Rn₎

1

χ2B Lu2(·)(Rn₎

.

For the estimate ofL22, we have|(x–y1,x–y2)|2n–α≈ |x–y1|2n–α. Using _p1

2(·) =

1

g(·)+ 1

u2(·),

λ1< –₂α_n, Lemmas2.3,2.6and the generalized Hölder inequality, we have

Tα,2

f1χ(2B)c,b₂(·) –{b₂}_Bf₂χ₂_B(x) ≤C

Rn

|f1(y1)χ(2B)c(y₁)||f₂(y₂)χ₂_B(y₂)||b₂(y₂) –{b₂}_B| |(x–y1,x–y2)|2n–α

dy1dy2

≤C

(2B)c

|f1(y1)|

|x–y1|2n–α

dy1

2B

f2(y2) b2(y2) –{b2}B dy2

≤C f2χ2B Lp2(·)₍_Rn₎b2–{b2}B

χ2B_Lp₂₍·)

(Rn₎ ∞

k=1

2k+1_B_\₂k_B

|f1(y1)|

|x–y1|2n–α

dy1

≤C f2χ2B Lp2(·)₍_Rn₎b2–{b2}B

χ2B_Lu2(·)₍_Rn₎ χ2B _Lg(·)₍_Rn₎ ×

∞

k=1

2k+1_B_\₂k_B

|f1(y1)|

|x–y1|2n–α

dy1

≤C f2 _B˙p₂₍·),λ2(Rn₎|2B|λ2 χ2B Lp2(·)(Rn₎ b2 CBMOu2(·),v2(Rn₎|2B|v2 χ2B Lu2(·)(Rn₎ × χ2B _Lg(·)₍_Rn₎

∞

k=1

2k–1R–2n+α

2k+1_B

f1(y1) dy1

≤C f2 _B˙p2(·),λ2₍_Rn₎|2B|λ2+v2+1 b2 _CBMOu2(·),v2₍_Rn₎ ×

∞

k=1

Lp1(·)₍_Rn₎ ≤C f2 _B˙p2(·),λ2(Rn₎ f1 _B˙p1(·),λ1(Rn₎|B|λ2+v2+1–2+α/n+1+λ1

× b2 CBMOu2(·),v2(Rn₎ ∞

k=1

2kn(–2+α/n+λ1+1)

≤C f1 _B˙p₁₍·),λ₁₍_Rn₎ f2 _B˙p₂₍·),λ₂₍_Rn₎|B|λ1+λ2+v2+α/n b2 CBMOu2(·),v2(Rn₎ ∞

k=1

2kn(2αn+λ1)

≤C|B|λ1+λ2+v2+α/n _b

2 _CBMOu2(·),v2₍_Rn₎

2

i=1

(14)

Thus, from _q1₍_·₎=_u1

1(·)+

1

q1(·) and Lemmas2.5,2.6, we get

L22≤C|B|λ1+λ2+v2+α/n b2 CBMOu2(·),v2(Rn₎

2

i=1

fi _B˙pi(·),λ_i₍_Rn₎b1–{b1}B

χB_Lq(·)₍_Rn₎ ≤C|B|λ1+λ2+v2+α/n _b

2 CBMOu2(·),v2(Rn₎b1–{b1}B

χB_Lu1(·)₍_Rn₎ × χB Lq1(·)(Rn₎

2

i=1

fi _B˙pi(·),λi(Rn₎ ≤C|B|λ1+λ2+v2+α/n _b

2 _CBMOu2(·),v2₍_Rn₎ b1 _CBMOu1(·),v1₍_Rn₎|B|v1 χB _Lu1(·)₍_Rn₎ × χB Lq1(·)(Rn₎

2

i=1

fi _B˙pi(·),λi(Rn₎

≤C|B|λ+1/q(·)

2

i=1

. (4.12)

ForL23, noticing that|(x–y1,x–y2)|2n≥ |x–y2|2nand_p1

2(·)=

1

g(·)+ 1

u2(·). By Lemmas2.3,

2.6,v2+λ2+α/n< 0, the generalized Hölder inequality and the Minkowski inequality, we

get

Tα,2

f1χ2B,

b2(·) –{b2}B

f2χ(2B)c(x) ≤C

Rn

|f1(y1)χ2B(y1)||f2(y2)χ(2B)c(y₂)||b₂(y₂) –{b₂}_B| |(x–y1,x–y2)|2n–α

dy1dy2

≤C

2B

f1(y1) dy1

(2B)c

|f2(y2)||b2(y2) –{b2}B|

|x–y2|2n–α

dy2

≤C f1χ2B _Lp₁₍·)₍_Rn₎ χ2B

Lp1(·)(Rn₎

(2B)c

|f2(y2)||b2(y2) –{b2}B|

|x–y2|2n–α

dy2

≤C|B|λ1+1 _f

1 _B˙p₁₍·),λ₁₍_Rn₎ _∞

k=1

2kB –2+α/n

×

2k+1_B

f2(y2) b2(y2) –{b2}2k+1_B+{b2}2k+1_B–{b2}B dy2

≤C|B|λ1+1 _f

1 _B˙p₁₍·),λ1(Rn₎ _∞

k=1

2kB –2+α/n f2χ2k+1_B _Lp₂₍·)₍_Rn₎

×b2–{b2}2k+1_B

χ₂k+1_B

Lp2(·)(Rn₎+ {b2}2k+1_B–{b2}B χ2k+1_B

Lp2(·)(Rn₎

≤C|B|λ1+1 _f

1 _B˙p₁₍·),λ1(Rn₎ _∞

k=1

k 2kB –2+α/n+λ2+v2

f2 _B˙p₂₍·),λ2(Rn₎

× χ₂k+1_B _Lp₂₍·)₍_Rn₎ b2 _CBMOu2(·),v2₍_Rn₎ χ₂k+1_B _Lu₂₍·)₍_Rn₎ χ₂k+1_B _Lg(·)₍_Rn₎

≤C|B|λ1+1–2+α/n+λ2+v2+1 _f

(15)

× b2 CBMOu2(·),v2(Rn₎ ∞

k=1

k2kn(–2+α/n+λ2+v2+1)

≤C f1 _B˙p1(·),λ1(Rn₎ f2 _B˙p2(·),λ2(Rn₎|B|λ1+λ2+v2+α/n b2 CBMOu2(·),v2(Rn₎ ∞

k=1

k2–kn

≤C|B|λ1+λ2+v2+α/n _b

2 CBMOu2(·),v2(Rn₎

2

i=1

fi _B˙pi(·),λi(Rn₎,

where

_{b2}2k+1_B–{b2}B ≤ k

j=0

_{b2}2j+1B–{b2}2jB

≤

k

j=0

1

|2j_B_|

2j_B

b2(y) –{b2}2j+1_B dy

≤C

k

j=0

1

|2j_B_|b2(·) –{b2}2j+1_B

χ₂j+1_B

Lu2(·)(Rn₎ χ2j+1_B

Lu2(·)(Rn₎

≤C

k

j=0

1

|2j_B|b2(·) –{b2}2j+1B

χ₂j+1_B

Lu2(·)₍_Rn₎

|2j+1_B_|

χ₂j+1_B _Lu2(·)₍_Rn₎

≤C b2 CBMOu2(·),v2(Rn₎

k

j=0

2j+1B v2 χ₂j+1_B _Lu2(·)₍_Rn₎

1

χ₂j+1_B _Lu2(·)₍_Rn₎ ≤C b2 CBMOu2(·),v2(Rn₎(k+ 1) 2k+1B

v2

,

forv2> 0.

Hence,

L23≤C|B|λ+1/q(·) 2

i=1

. (4.13)

ForL24, using Lemmas2.3,2.6,v2+λ2+α/n< 0 and the generalized Hölder inequality, we

obtain

_T_α_,2

f1χ(2B)c,b₂(·) –{b₂}_Bf₂χ₍₂_B₎c(x) ≤C

Rn

|f1(y1)χ(2B)c(y₁)||f₂(y₂)χ₍₂_B₎c(y₂)||b₂(y₂) –{b₂}_B| |(x–y1,x–y2)|2n–α

dy1dy2

≤C

(2B)c

|f1(y1)|

|x–y1|n–α/2

dy1

(2B)c

|f2(y2)||b2(y2) –{b2}B|

|x–y2|n–α/2

dy2

≤C

∞

k1=1

2k1_B –1+2αn

2k1+1_B

f1(y1) dy1

× ∞

k2=1

2k2_B –1+2αn

2k2+1B