SUN Ai-wen,WANG Min,SHU Li-sheng

(School of Mathematics and Computer Science,Anhui Normal University,Wuhu 241003,China)

LIPSCHITZ TYPE SMOOTHNESS OF MULTILINEAR FRACTIONAL INTEGRAL ON VARIABLE EXPONENTS SPACES

SUN Ai-wen,WANG Min,SHU Li-sheng

(School of Mathematics and Computer Science,Anhui Normal University,Wuhu 241003,China)

In this paper,we study the boundedness of multilinear fractionalintegraloperators on variable exponent spaces.It is obtained that these operators are both bounded from strong and weak Lebesgue spaces with variable exponent spaces into Lipschitz type spaces with variable exponent,which gives some new results for previous published papers.A simple way is obtained that is colsely linked with a class of fractional integral operator.

Lipschitz spaces;multilinear fractional integral;variable exponent

1 Introduction

Let(Rn)m=Rn× Rn× ···× Rnbe the m-fold product space of Rn,the multilinear fractionalintegrals on Rnare defined by

When The famous Hardy-Littlewood-Sobolev theorem tells us that the fractional integral operator Iβis a bounded operator from the usual Lebesgue spaces Lp1(Rn)to Lp2(Rn)when 1 ＜ p1＜ p2＜ ∞ andp1

1?Kenig and Stein[1]as wellas Grafakos and Kalton[2]considered the boundedness of a family of related multilinear fractional integrals.Lan[3]presented the boundedness of multilinear fractional integral operators on weak type Hardy spaces.Recently,Yasuo[4] considered the boundedness of multilinear fractionalintegraloperators on Herz spaces.

It is well known that function spaces with variable exponents were intensively studied during the past 20 years,due to their applications to PDE with non-standard growth conditions and so on,we mention e.g.[5,6].A great dealofwork was done to extend the theory offractionalintegralon the classical Lebesgue spaces to the variable exponent case(see[7–9]). However,these articles do not consider the behavior of Iβwhen.Recently,Ramseyer,Salinas and Viviani[10]studied that Lipschitz type smoothness of fractional integral on variable exponent spaces,when.Hence,whenit will be an interesting problem whether we can establish the boundedness of multilinear fractional integral from Lebesgue spaces Lp(·)into Lipschitz-type spaces with variable exponents.The main purpose of this paper is to answer the above problem.

To meet the requirements in the next sections,here,the basic elements ofthe theory of the Lebsegue spaces with variable exponent are briefly presented.

Lp(·)(?)is a Banach space with the norm defined by

We say a function p(·):Rn?→ R is locally log-H¨older continuous,if there exists a constant C such that

For brevity,C always means a positive constant independent of the main parameters and may change from one occurrence to another.be the characteristic function of the setdenotes the Lebesgue measure of S.f ～ g means C?1g ≤ f ≤ C g.

Defi nition 1.1[10]Given an exponent function p(·)we say that a measurable function f belongs to Lp(·),∞if there exists a constant C such that for every t ＞ 0.

It is not diffi cult to see that

is a quasi-norm in Lp(·),∞.

Defi nition 1.2[10]Given 0 ＜ β ＜ n and an exponent function p(·)with 1 ＜ p?≤p+＜ ∞ we say that a locally integrable function f belongs to Lipβ,p(·)if there exists a constant C such that

RemarkIt is easy to see that in definition the average can be replaced by a constant in the following sense

In this paper,we consider the case of bilinear fractional integral.

Defi nition 1.3[4]

where 0 ＜ β ＜ 2n.

Now it is in this position to state our results.

2 Lemmas

Lemma 2.1[11]If p(·) ∈ P(Rn),then for all f ∈ Lp(·)(Rn)and all g ∈ Lp′(·)(Rn)we have

where rp:=1+1/p?? 1/p+.

Lemma 2.2[10]Let p(·)be an exponent function in Plog(Rn)such that 1 ＜ p?≤ p+＜∞ and p(x) ≤ p(∞)for|x|＞ r0with r0＞ 1.Then there exiZsts a positive constant C depending on r0and the constants associated Plog(Rn)such thatfor every ball B and f ∈ Lp(·),∞.

The following lemma see Corollary 4.5.9 in[12].

and

Lemma 2.3Let p(·) ∈ Plog(Rn),then for every ball B ? Rn,we have

We remark that Lemma 2.4 were showed in[13]and we willgive the proof of it.

Lemma 2.4Let p(·) ∈ Plog(Rn)and x2∈ 2B(x1,r),then we have

ProofWe consider two cases,by Lemma 2.3.

Case 1|B|≥ 1.

Case 2|B|≤ 1.

where we denote that x′∈ B(x1,r)and x′′∈ B(x2,r).

Indeed,since x2∈ 2B(x1,r),x′∈ B(x1,r)and x′′∈ B(x2,r)we note that|x′? x′| ≤ 4r, we make use of local-H¨older continuity of p(x)and get,

Lemma 2.5Let pi(·) ∈ Plog(Rn)for i=1,2 and,then for every

ball B=B(x,r) ? Rn,we have

ProofWe will give the proof of inequality(2.2),the argument for inequality(2.1)is similar,we omit the details here.We consider two cases,by Lemma 2.3.

Case 1|B|≤ 1.

Case 2|B|≥ 1.

Lemma 2.6Let p(·) ∈ Plog(Rn),then there exists a constant C ＞ 0 such that for all balls B and allmeasurable subsets S=B(x0,r0) ? B=B(x1,r1),

RemarkWe can easily show that inequality(2.4)implies ‖χ2B‖p′(·)≤ C‖χB‖p′(·).

ProofWe will prove inequality(2.3),the argument for inequality(2.4)is similar,we omit the details here.We consider three cases,by Lemma 2.3.

where we denote that xS∈ S and xB∈ B.

Indeed,since|xB? xS|≤ 2r1,we make use of local-H¨older continuity of p′(x)and get

Lemma 2.7Let p(·) ∈ Plog(Rn)such that 1 ＜ p?≤ p+＜ ∞,B=B(x0,R)and k ＜ n ? n/p?,then there exists a constant C ＞ 0 such that

ProofUsing Lemma 2.1,we obtain

Lemma 2.6 gives

Lemma 2.8Let p(·) ∈ Plog(Rn)such that 1 ＜ p?≤ p+＜ ∞,B=B(x0,R)and k ＞ n ? n/p+,then there exists a constant C ＞ 0 such that

ProofApplying Lemma 2.1,we derive the estimate

Lemma 2.6 implies that

Lemma 2.9Suppose pi(·)then

ProofFor y1,y2∈ 2B,one can obtain the following inequality in[4]

When n ＜ β ＜ 2n,using Lemma 2.1 and 2.5,we obtain

When 0 ＜ β ＜ n,we write

when j ＞ ?1 we define D3=0.

For y1∈ Aiy2∈ Aj,we have|y1? y2|≥ |y1|? |y2|＞ 2i?2R.Then

Now Lemma 2.1 yields

By Lemma 2.5,we get

Next we estimate D2.

Noting that|y1? y2|≤ |y1|+|y2|＜ C2iR for y1∈ Ai,y2∈ Aj,using Lemmas 2.1 and 2.7,we have

By Lemmas 2.4 and 2.5,we arrive at the inequality

Finally,we estimate D3.

We note y1∈ Ai,y2∈ Aj,|y1? y2|≥ |y1|? |y2|＞ 2j?2R and derive

Hence,we apply Lemma 2.1 and 2.5 and obtain

When β =n,the proof is similar.Therefore we omit the details.We use the following inequality

As long as we change the conclusion of Lemma 2.1 into the conclusion of Lemma 2.2 in the proof of Lemmas 2.7–2.9,we can obtain the corresponding conclusions in Lp(·),∞space.

Corollary 2.1Let p(·) ∈ Plog(Rn)such that 1 ＜ p?≤ p+＜ ∞,B=B(x0,R), p(x) ≤ p(∞)for|x|＞ r0with r0＞ 1 and k ＜ n ? n/p?,then there exists a constant C ＞ 0 such that Z

Corollary 2.2Let p(·) ∈ Plog(Rn)such that 1 ＜ p?≤ p+＜ ∞,B=B(x0,R), p(x) ≤ p(∞)for|x|＞ r0with r0＞ 1 and k ＞ n ? n/p+,then there exists a constant C ＞ 0 such that Z

Corollary 2.3Let pi(·) ∈ Plog(Rn)such that 1 ＜ p?i≤p+i＜ ∞ ,pi(x) ≤ pi(∞)for i=1,2 and|x|＞ r0with r0＞ 1.Suppose B=B(x0,R)andthen

3 Proof of Theorems

We will give the proof of the Theorem 1.1 below.In Corollary 2.1–Corollary 2.3,we obtain the corresponding results in Lp(·),∞space.The argument for Theorem 1.2 is similar, we omit the details here.

Proof of Theorem 1.1We write

And we need to estimate four terms ?Iβ(f1χ2B,f2χ2B),?Iβ(f1χRn2B,f2χ2B),?Iβ(f1χ2B,f2χRn2B) and ?Iβ(f1χRn2B,f2χRn2B).

Hence,we arrive at the inequality

Next we estimate ?Iβ(f1χRn2B,f2χ2B)and ?Iβ(f1χ2B,f2χRn2B).

We only estimate ?Iβ(f1χRn2B,f2χ2B)and the estimate for ?Iβ(f1χ2B,f2χRn2B)is similar,we omit the details here.

Let c= ?Iβ(f1χRn2B,f2χ2B)(x0),then for x ∈ B,we have

Applying Lemma 2.8,2.1 and 2.5,we obtain

Thus we get that

Finally we estimate I?β(f1χRn2B,f2χRn2B).

By Lemmas 2.8 and 2.5,we havewhere we can take s1and s2such that s1＜ n/p+1,s2＜ n/p+2and s1+s2= β ? 1.

Hence,we obtain

Consequently we prove Theorem 1.1.

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變指數(shù)空間上多線性分?jǐn)?shù)次積分的 Lipschitz 光滑性

孫愛(ài)文,王 敏,束立生
(安徽師范大學(xué)數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽蕪湖 241003)

本文研究了多線性分?jǐn)?shù)次積分算子在變指數(shù)空間的有界性.利用多線性分?jǐn)?shù)次積分轉(zhuǎn)化為相對(duì)應(yīng)的分?jǐn)?shù)次積分的方法, 獲得了它從變指數(shù)強(qiáng)和弱Lebesgue空間到變指數(shù)Lipschitz空間的有界性, 推廣了先前的研究結(jié)果.

Lipschitz空間; 多線性分?jǐn)?shù)次積分; 變指數(shù)

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A < class="emphasis_bold">Article ID:0255-7797(2017)02-0315-10

0255-7797(2017)02-0315-10

?Received date:2015-09-09 Accepted date:2016-04-27

Foundation item:Supported by National Natural Science Foundation of China(11201003; 11301006);and University NSR Pro ject of Anhui Province(KJ2015A117;KJ2014A087).

Biography:Sun Aiwen(1982–),female,born at Bengbu,Anhui,associate professor,major in harmonic analysis.