Boundedness of Vector-Valued Multilinear Singular Integral Operators on Generalized Morrey Spaces

(School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China)



Boundedness of Vector-Valued Multilinear Singular Integral Operators on Generalized Morrey Spaces

YUFei

(School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China)

AbstractIn this paper, we mainly investigate the boundedness of vector-valued multilinear singular integral operators on generalized Morrey spaces.

Key wordssingular integral operator; vector-valued multilinear singular integral operator; BMO; generalized Morrey spaces

The multilinear singular integral operatorTAwas first introduced by Cohen and Gosselin, which is defined as follows:

TheLp(p>1)boundednessofthemultilinearsingularintegraloperatorisprovedbytheauthorsof[1-3].Later,HuandYangprovedavariantsharpestimateforthemultilinearsingularintegraloperatorsin[4].In2010,Liuconsideredthemultilinearsingularintegraloperatorsonclassicalmorreyspacein[5].Recently,DuandHuangstudiedtheboundednessofvector-valuedmultilinearsingularintegraloperatoronvariableexponentLebesguespacesin[6].Thevector-valuedmultilinearsingularintegraloperatorisdefinedasfollws:

Whenλ=0, Lp,0(Rn)=Lp(Rn). Whenλ=n, Lp,n(Rn)=L∞(Rn). Ifλ<0 orλ>n, then Lp,λ={0}. The generalized Morrey spacesMr,φ(Rn) were first defined by Guliyev in [12]. The generalized Morrey spaces recover the classical Morrey spaces, which will be explained in next section.

1Preliminaries

Inthissection,wewillgivesomebasicdefinitionsandlemmas,whichwillbeusedintheproofofourmainresults.

Definition1.1Fixε>0.LetSandS′beSchwartzspaceanditsdual, T:S→S′bealinearoperator.IfthereexistsalocallyintegrabalfunctionK(x,y)onRn×Rn{(x,y)∈Rn×R:x=y}suchthatT(f)(x)=∫RnK(x,y)f(y)dy,foreveryboundedandcompactlysupportedfunctionf,whereKsatises|K(x,y)|≤C|x-y|-nand|K(y,x)-K(z,x)|+|K(x,y)-K(x,z)|≤C|y-z|ε|x-z|-n-ε,if2|y-z|≤|x-z|.ThroughoutthepaperCwilldenoteapositiveconstantwhichmaybedifferentfromlinetoline.

Definition 1.2Letmjbe positive integers (j=1…,l),m1+…+ml=m, andAjbe functions on Rn(j=1,…,l). For 1

Definition 1.4[10]We call functionΦ(t) a Young function, if functionΦ(t) is a contious, nonnegative, strictly increasing and convex function on [0,∞) withΦ(0)=0 andΦ(t)→∞. TheΦ-average of a functionfover a cubeQis defined as

In the following, we give some lemmas which will play important roles in proof of our main results.

Lemma 1.1[6]Let 1

Remark 1. This Lemma can get from Theorem 2 in [6].

Lemma 1.3[3]LetAbe a function on Rnand DαA∈Λq(Rn) for allαwith |α|=mandq>n. Then

Lemma1.4[14](1)Forall1≤p<∞,thefollowingistrue

(2)Letb∈BMO(Rn). Then there exists a constantC>0 such that

for 0<2r≤t.

and

Remark2.Ifweusetheballinsteadofcube,theaboveresultsstillhold.

2Mainresultanditsproof

Theorem2.1Let1

then|TA|sisboundedfromMp,φ1(Rn) toMp,φ2(Rn) for all ‖|f|s‖Lp(Rn)<∞.

ForI, by Lemma 2.1 then we have

On the other hand,

Then we can get

Now let us estimateII

I1+I2+I3+I4

We first estimateI1. By Lemma 2.3 and 2.4, we have

So we get

Then

Using the same method in proof ofI2, we can get

According to the above estimate, we obtain

Thus,

Then according to the condition,

The proof is completed.

References:

[1]COHENJ.AsharpestimateforamultilinearsingularintegralinRn[J].IndianaUnivMathJ, 1981,30(5):693-702.

[2]COHENJ,GOSSELINJ.OnmultilinearsingularintegralsonRn[J].StudiaMath, 1982,72(4):199-223.

[3]COHENJ,GOSSELINJ.ABMOestimateformultilinearsingularintegrals[J].IllinoisJMath, 1986,30(3):445-464.

[4]HUG,YANGD.Avariantsharpestimateformultilinearsingularintegraloperators[J].StudiaMath, 2000,141(1):22-25.

[5]LIUL.BoundednessformultilinearsingularintegraloperatorsonMorreyspaces[J].BullMalaysMathSciSoc, 2010,33(1):93-103.

[6]DUJ,HUANGC,LIUL.Boundednessforvector-valuedmultilinearsingularintegraloperatoronLpspaceswithvariableexponent[J].BullAcadStiinteRepubMoldMath, 2012,3(70):3-15.

[7]PREZC,TRUJILLO-GONZALEZR.Sharpweightedestimatesformultilinearcommutators[J].LondMathSoc, 2002,65(03):672-692.

[8]PREZC,PRADOLINIG.Sharpweightedendpointestimatesforcommutatorsofsingularintegrals[J].MichiganMathJ, 2001,49(1):23-37.

[9]PREZC,TRUJILLO-GONZLEZR.Sharpweightedestimatesforvector-valuedsingularintegraloperatorsandcommutators[J].TohokuMathJ, 2003,55(1):109-129.

[10]PEREZC.Endpointestimatesforcommutatorsofsingularintegraloperators[J].JFunctAnal, 1995,128(1):163-185.

[11]MORREYJRCB.Onthesolutionsofquasi-linearellipticpartialdierentialequations[J].TransAmMathSoc, 1938,38(1):126-166.

[12]GULIYEVVS,ALIYEVSS,KARAMANT.BoundednessofsublinearoperatorsandcommutatorsongeneralizedMorreyspaces[J].IntegrEquOperTheor, 2011,71(3):327-355.

[13]KARLOVICHA,LERNERA.CommutatorsofsingularintegralsongeneralizedLpspaceswithvariableexponent[J].PublMath, 2005,49(1):111-125.

[14]GRAFAKOSL.ClassicalandmodernFourieranalysis[M].NewJersey:PrenticeHall, 2004.

(编辑胡文杰)

向量值多线性奇异积分算子在广义Morrey空间上的有界性

俞飞*

(安徽师范大学数学计算机科学学院，中国 芜湖241000)

摘要本文主要讨论向量值多线性奇异积分算子在广义Morrey空间上的有界性.

关键词奇异积分算子；向量值多线性奇异积分算子；有界平均振动空间；广义Morrey空间

中图分类号O174.2

文献标识码A

文章编号1000-2537(2015)05-0076-08

通讯作者*,E-mail：yf2014620@sina.com

基金项目：This paper was supported by the National Nature Science Foundation of China (No.11201003) and NNSF (No.KJ2012A133) of Anhui Province in China

收稿日期：2014-06-30

DOI:10.7612/j.issn.1000-2537.2015.05.013