非齐型空间上极大交换子的端点估计

GuoenHuDepartmentofAppliedMathematics,UniversityofInformationEngineering,Zhengzhou450002,People'sRepublicofChinaandYanMeng,DachunYangySchoolofMathematicalSciences,BeijingNormalUniversity,Beijing100875,People'sRepublicofChinaAbstractCertainweaktypeendpointestimatesareestablishedforthemaximalcommutatorsgeneratedbyCalderon-ZygmundoperatorsandOscexpLr()functionsforr1undertheconditionthattheunderlyingmeasureonlysatisessomegrowthcondition,wherethekernelsofCalderon-ZgymundoperatorsonlysatisfythestandardsizeconditionandsomeHormandertypesmoothcondition,andOscexpLr()arethespacesofOrlicztypesatisfyingthatOscexpLr()=RBMO()ifr=1andOscexpLr()RBMO()ifr1.More-over,thesameweaktypeendpointestimatesforthemaximaloperatorsassociatedwiththeHardy-LittlewoodmaximalfunctionandOscexpLr()functionswithr1arealsopre-sented.Keywords:Calderon-Zygmundoperator,Orliczspace,maximalcommutator,Hardy-Littlewoodmaximalfunction,endpointestimate.2000MathematicsSubjectClassication:47B4742B2043A99.ThisprojectwassupportedbyNNSF(No.10271015)ofChinaandthethirdauthorwasalsosupportedbyRFDP(No.20020027004)ofChina.yCorrespondingauthor.Tel.:[phone]Emailaddress:[e-mail].1issaidtosatisfythedoublingconditionifthereisapositiveconstantCsuchthat(B(x;2r))C(B(x;r))forallx2suppandr0,whereB(x;r)=fy2Rd:jyxjrg.However,duringthelastseveralyears,manyclassicalresultshavebeenprovedstillvalidiftheunderlyingmeasureisaRadonmeasureonRdwhichonlysatisesthefollowinggrowthconditionthatthereexistsaconstantC0suchthat(B(x;r))Crn(1:1)forallx2Rdandr0,wherenisaxednumbersatisfying0nd;see[5,4,6,8,9,10,12].TheEuclideanspaceRdequippedwithaRadonmeasurethatonlysatises(1.1)iscalledanon-homogeneousspacesincemaybenotdoubling.Themotivationfordevelopingtheanalysisonnon-homogeneousspacesandsomeexamplesofnon-doublingmeasurescanbefoundin[14].Weonlypointoutthattheanalysisonnon-homogeneousspacesplayedanessentialroleinsolvingthelongopenPainleve'sproblembyTolsain[13].ThepurposeofthispaperistoestablishsomeweaktypeendpointestimatesforthemaximalcommutatorsgeneratedbyCalderon-Zygmundoperators,whosekernelssatisfythestandardsizeconditionandsomeweakersmoothcondition,withOscexpLr()functions,wherer1.WealsoprovethatthecommutatorsassociatedwiththeHardy-LittlewoodmaximalfunctionsandOscexpLr()functionsforr1satisfythesameweaktypeendpointestimates.Beforestatingourresults,werstrecallsomenecessarynotationanddenitions.Throughoutthispaper,anycubeisaclosedcubeinRdwithsidesparalleltotheaxesandcenteredatsomepointofsupp().Wedenoteitssidelengthbyl(Q)anditscenterbyxQ.Letandbepositiveconstantssuchthat1andn.ForacubeQ,wesaythatQis(;)-doublingif(Q)(Q),whereQdenotesthecubeconcentricwithQandhavingsidelengthl(Q).FortwocubesQ1Q2,setKQ1;Q2=1+NQ1;Q2Xk=1(2kQ1)l(2kQ1)n;whereNQ1;Q2istherstpositiveintegerksuchthatl(2kQ1)l(Q2);see[8]forsomebasicpropertiesofKQ1;Q2.Denition1Forr1,alocallyintegrablefunctionfissaidtobelongtothespaceOscexpLr()ifthereisaconstantC10suchthat(i)foranyQ,fmeQ(f)expLr;Q=inf(0:1(2Q)ZQexpjfmeQ(f)j!rd2)C1;(ii)foranydoublingcubesQ1Q2,jmQ1(f)mQ2(f)jC1KQ1;Q2;whereforanycubeQinRd,eQisthesmallestdoublingcubecontainingQandhavingthesamecenterasQ,andmQ(f)isthemeanvalueoffonQ,namely,mQ(f)=1(Q)ZQf(x)d(x):TheminimalconstantC1satisfying(i)and(ii)isdenedtobetheOscexpLr()normoffanddenotedbykfkOscexpLr().ThedenitionofthespaceOscexpLr()isananalogyoftheclassicalOscexpLr(Rd)spacewhichwasintroducedbyPerezandTrujillo-Gonzalezin[7].Obviously,foranyr2r11,OscexpLr2()OscexpLr1()RBMO().Moreover,fromtheJohn-Nirenberginequalityinnon-homogeneousspaces(seeTheorem3.1in[8]),itfollowsthatOscexpL1()isjustthespaceRBMO()ofTolsain[8].LetKbeafunctiononRdRdnf(x;y):x=ygsatisfyingthatforx6=y,jK(x;y)jCjxyjn;(1:2)andZjxyj2jyy0jjK(x;y)K(x;y0)j+jK(y;x)K(y0;x)j d(x)C;(1:3)whereC0isapositiveconstant.For0,denethetruncatedoperatorsTbyT(f)(x)=ZjxyjK(x;y)f(y)d(y):(1:4)ItiswellknownthatiftheoperatorsTareboundedonL2()uniformlyon0,thenthereisanoperatorTwhichistheweaklimitas!0ofsomesubsequenceoftheuniformlyboundedoperatorsT.TheoperatorTisalsoboundedonL2()andsatisesthatforf2L2()withsuppf6=Rd,T(f)(x)=ZRdK(x;y)f(y)d(y);a:e:Rdnsuppf:Tolsa[8]introducedthespaceRBMO(),ananalogyoftheclassicalBMOspace,andestablishedtheLp()boundednessofthecommutatorofTandanyRBMO()functionb,whichisdenedbyTb(f)(x)=b(x)T(f)(x)T(bf)(x):(1:5)TolsaprovedthatifKsatisesthesizecondition(1.2)andthestandardregularityconditionthatforjxyj2jyy0j,jK(x;y)K(x;y0)j+jK(y;x)K(y0;x)jCjyy0jjxyjn+(1:6)1,andifTisboundedonL2(),thenforanyb2RBMO(),TbisboundedonLp()withboundCkbkRBMO()foranyp2(1;1).Recently,HuandLian