参数型Marcinkiewicz积分交换子在Hardy空间的有界性.pdf

第30卷第4期 新疆大学学报(自然科学版) V 1．30．N0．4 2013年 11月 JournalofXi~iangUniversity(NaturalScienceEdition) NOV．，2013 BoundednessofHigherOrderCommutatorsforthe ParametricM arcinkiewiczIntegralonHardySpace木 WU Rui—min。JIANGYin—sheng十 (CollegeofMathematicsandSystemSciences,XinjiangUniversity,Urumqi，Xinjiang83004~China) Abstract： TheauthorsstudythehigherordercommutatorsofparametricMracinkiewiczintegralsandprovethatthey areboundedfrom H l( )toL1( )withb∈BMO． 口 Keywords：Hradyspace，commutators，praametricMracinkiewiczintegral CLCnumber：O174．2 DocumentCode：A ArticleID ：1000—2839(2013)04—0418-04 参数型Marcinkiewicz积分交换子在Hardy空间的有界性 吴芮民，江寅生 (新疆大学数学与系统科学学院，新疆 乌鲁木齐 830046) 摘 要 ： 研究了参数型Marcinkiewicz积分高阶交换子并证明了其从Hl()到 L()有界，其中b∈BMO 关键词 ： Hrady空间，交换子，参数型Marcinkiewicz积分 0 Introduction LetTbeaCalder6n—Zygmundsingularintegraloperator．Foralocallyintegralfunctionbon ．thecommutator [b，T】ofTisdefinedby [b，Tlf(x)=易()(-厂)()一 (，)()． ItiSwel1knownthattheCalder6n．ZygmundsingularintegraloperatorTiSboundedfrom H (R)toL ( )．However, itwasobservedin[1】thatthecorrespondingresultfor[b，T】iSfalsewhenbiSaBMO function．In1995，P6rezin 1【]introducedasubspaceH1( )，andprovedthat[b，T】isboundedoperatorfromH1( )toL (R)ofrb∈BMD． Followingthedefinitionof2【]，in1998，Alvarezdefinedtheatomicspace ( )，ofr0P1，andprovedthatthe commutator[b，T】iSalsoboundedrfom ( )toLP(R)，ofrn／(n+1)P lin[3】． Beforestatingourresults，letUSfirstgivethedefinitionsandsomeknownresultsOff and bm ． SupposethatS istheunitspherein (n 2)equippedwiththenormalizedLebesguemeasuredo-：d-o(Z)．LetQbeahomogeneous fu