BesovspacesBp,q(Rn)andTriebel-LizorkinspacesFp,q(Rn)arerepresentingnowadayswellcently,Triebel[1]introducedtwonewscalesofspacesdenotedbyLrBp,q(Rn)andLrFp,q(Rn),基于Morrey空间的某些光滑空间的一致性袁文1,2,WINFRIEDSICKEL2杨大春1(1.北京师范大学数学科学学院,北京邮编100875;2.德国耶拿Friedrich-Schiller大学,耶拿邮编07743)摘要：在本文中,作者比较了HansTriebel最近引进的相关于Morrey-Campanato空间的光滑空间与Besov型和Triebel-Lizorkin型函数空间.这两类空间在近年引进并用以采用Morrey空间测量函数和分布的光滑性.关键词：Besov空间,Triebel-Lizorkin空间,Morrey-Campanato空间,小波,原子中图分类号：46E35,42C40OntheCoincidenceofCertainApproachestoSmoothnessSpacesRelatedtoMorreySpacesYUANWEN1,2,WINFRIEDSICKEL2,YANGDACHUN1(1.SchoolofMathematicalSciences,BeijingNormalUniversity,Beijing100875;2.MathematischesInstitut,Friedrich-Schiller-Universit¨atJena,07743Jena,Germany)Abstract:Inthispaper,wecomparetherecentapproachofHansTriebeltointroducesmoothnessspacesrelatedtoMorrey-CampanatospaceswithBesovtypeandTriebel-Lizorkintypespaces.ThesetwoscaleshavebeenintroducedsomeyearsagoandrepresentafurthervarianttomeasuresmoothnessbyusingMorreyspaces.Keywords:Besovspace,Triebel-Lizorkinspace,Morrey-Campanatospace,wavelet,atom0Introductionandmainresultsssacceptedconceptstodescribetheregularity(smoothness)offunctionsanddistributions.Re-ssrespectively.Thenewparameterrisconnectedwithlocalapproximationpropertiesbycom-pactlysupportedwavelets.Forthisreasonhecalledtheclasseslocalspaces.InspiritthisisveryclosetoCampanatospaceswhicharebasedonlocalapproximationpropertiesbypoly-nomialsofacertaindegree.However,hismainaimwasconsistinginanapplicationofthesespacesinconnectionwiththeheatequationandtheNavier-Stokesequation;seeChapters5and6ofthequotedbook[1].基金项目：DachunYangissupportedbytheNationalNaturalScienceFoundationofChina(GrantNo.11171027),andWenYuanbytheAlexandervonHumboldtFoundationandtheNationalNaturalScienceFoun-dationofChina(GrantNo.11101038).BothDachunYangandWenYuanarealsopartiallysupportedbytheSpecializedResearchFundfortheDoctoralProgramofHigherEducationofChina(GrandNo.20120003110003).作者简介：Correspondenceauthor:YangDachun(1963-),male,professor,majorresearchdirection:functionspaces.-1-Bp,qτ(Rn)andTriebel-LizorkintypeFp,qτ(Rn)spaces.WhereaswearenotsureabouttheaimsofElBaraka[2,3,4],whojustconsideredthespacesBp,qτ(Rn)(p,q∈[1,∞])foritsown,forQαspaces.ForafirstsystematicstudyofthescalesBp,qτ(Rn)andFp,qτ(Rn)werefertothediscussionoftheirdefinitionsaswellastheirrelationstoBesov-MorreyspacesNu,p,q(Rn)andTriebel-Lizorkin-MorreyspacesEu,p,q(Rn).LuniformspacesAs,p,q,unif(Rn)(incase,thereisnoreasontodistinguishbetweenB-andF-spacesIncaseofthespacesAs,p,qτ(Rn)itiseasilyseenthatthespaceAs,p,q,unif(Rn)isindependentLn(τ−1/p)Asp,q(Rn)=As,p,q,unif(Rn),TotallydifferentintentionshavebeenleadtotheintroductionoftheBesovtypespacess,s,s,theintentionsoftwooftheauthors(see[5,6]),wereconsistinginclarifyingtherelationofBesovandTriebel-LizorkinspacestoQαspaces,representingarelativelynewdevelopmentinharmonicanalysis.Werefer,e.g.,toDafniandXiao[7]andEss´en,Janson,PengandXiao[8]s,s,lecturenoteoftheauthors[9].Furthermorewerefertotherecentsurvey[10]foradetailedssThemainpurposeofthispaperistostudytherelationsbetweenTriebel’slocalspacesrAsp,q(Rn)andtheBesov-typeandTriebel-Lizorkin-typespacesAs,p,qτ(Rn)andtheirassociatedτweshallsimplyr...