n维Hardy算子的一些新进展.pdf

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1、第42卷第6期数学进展Vo1．42．NO．62013年12月ADVANCESINMATHEMATICSDec．，2013SomeRecentProgressof．dimensionalHardyOperatorsLUShanzhen(SchoolMathematicalSciences，BeringNormalUniversity，Beijing，100875，P．R．China)Abstract：ThisiSasurveyonn—dinlensionalHardyoperatorsbasedontheresea∞rc∑hw一orkoftheauthorandhiscoop

2、eratorsinrecentfiveyearsandbasedonthelectureoftheplenarytalkgivenbytheauthorattheInternati0nalConferenceonHarmonicAnalysisandApplicationsatNanjing(May20—25，20121．Keywords：Hardyoperator；sharpbound；commutator；functionspaceMR(2000)SubjectClassification：42B25；42B35；41A44／cLCnumber：O174．2Docume

3、ntCOde：AArticleID：1000—0917(2013106—0737—110IntroductionItiswellknownthataveragingoperatorsplayanimportantroleinharmonicanalysis．Forexample，theHardy—Littlewoodmaximaloperatorcontrolsmanykindsofoperators∞i∑nan一alysis．ThereforejthestudyontheHardy—Littlewoodmaximaloperatorisverymeaningfulandh

4、asbeenfullydiscussed．However，mostofthestudyforHardyoperatorsjanotherkindofaveragingoperators，areonlyinthecaseofone—dimension．Webeginourdiscusswiththecaseofonevariable．In1920，inspiredbythediscretein—equality(wheream0，P>1，Hardy[。】firstsetupthefollowinginequality。。(掣)()。。州where1

5、dtand，0．FurthermoreI()isthebestconstantTheclassicalHardyoperatorforone—dimensionisdefinedbyHf(=：xTheresultin【20]showsthat日llLp(R)—斗Lp(豫)pP——1Receiveddate：2012—07-29．Foundationitem：ThisworkissupportedbytheKeyLaboratoryonMathematicsandComplexSystem(atBeijingNormalUniversity)fromtheMinistryof

6、Education，China，alsosupportedbyNSFC(No．10931001)．E-mail：lusz@bfil1．edu．cn738数学进展42卷ThestudyforHardyoperatorshasdrawnconsiderableattentionandalargenumberofpapershaveappearedwhichdealwithitsalterna．tiveproo~，variousgeneralizat]onsandapplications．KUfnerandPersson[23Jwroteabookabouttheoperator

7、f0．31in2003．Asahigherdimensionalgeneralization．Faris[6Jfirstdefinedn—dimensionalHardyoperatorsin1976duringhisstudyofquantummechanics．In1995．ChristandGrafakos[3Jgavethefollowingequivalentdefinitionofn—dimensionalHardyoperators：巩(1，)(z)=1unfzf札f(y)dy，X∈＼{@)Herel