Isometric classification of norms in rearrangement

arXiv:math/9512206v1[math.FA]12Dec1995ISOMETRICCLASSIFICATIONOFNORMSINREARRANGEMENT-INVARIANTFUNCTIONSPACESBEATARANDRIANANTOANINAAbstract.Supposethatarealnonatomicfunctionspaceon[0,1]isequippedwithtworearrange-ment-invariantnormsk·kand|||·|||.Westudythequestionwhetherornotthefactthat(X,k·k)isisometricto(X,|||·|||)impliesthatkfk=|||f|||forallfinX.WeshowthatinstrictlymonotoneOrliczandLorentzspacesthisisequivalenttoaskingwhetherornotthenormsaredefinedbyequalOrliczfunctions,resp.Lorentzweights.Weshowthattheaboveimplicationholdstrueinmostrearrangement-invariantspaces,butwealsoidentifyaclassofOrliczspaceswhereitfails.WeprovideacompletedescriptionofOrliczfunctionsϕ6=ψwiththepropertythatLϕandLψareisometric.1.IntroductionInthispaperwestudythefollowingquestion:Question1.Supposethatanonatomicfunctionspaceon[0,1]isequippedwithtworearrange-ment-invariantnormsk·kand|||·|||suchthat(X,k·k)and(X,|||·|||)areisometric.Doesitimplythatthenormsk·kand|||·|||aresame?Heretheword“same”couldbeunderstoodintwoways:(a)wecouldsaythatk·kand|||·|||aresameifkfk=|||f|||forallfinX,i.e.iftheidentitymapId:(X,k·k)−→(X,|||·|||)isanisometry,or(b)ifbothk·kand|||·|||areOrliczorLorentznormswecouldsaythattheyaresameiftheyaredefinedbyequalOrliczfunctionsorLorentzweights,respectively.Itiswell-knownthatifwedonotrequire|||·|||toberearrangement-invariantthentheanswertoQuestion1isnoineitherthesense(a)or(b),evenwhen(X,k·k)=Lp[0,1]withtheusualnorm(seee.g.[9]).Question1(a)hasbeenpreviouslystudiedwiththeadditionalassumptionthat(X,k·k)=(Lp[0,1],k·kp)(see[2,1,3,5]).Question1(b)wasaskedbyS.DilworthandH.Hudzik.Somewhattoauthor’ssurprisetheanswertobothQuestion1(a)and1(b)isnegative–thereexistOrliczfunctionsϕ6=ψ(then,clearly,k·kϕ,k·kψaredifferentalsointhesence(a))sothatLϕandLψareisometric.1991MathematicsSubjectClassification.46B,46E.Keywordsandphrases.isometries,rearrangement-invariantfunctionspaces,Orliczspaces,Lorentzspaces.1Infact,Question1(a)and1(b)areequivalentforstrictlymonotoneOrlicz-LorentzspacesX,Y,i.e.Id:X−→YisanisometryifandonlyiftheOrliczfunctionsϕX,ϕYandLorentzweightswX,wYcoincide(Theorem7;thisisnottrueingeneral,see[5]).Tooursurprisetheproofismuchlessobviousthanonemightexpect.Question1(a)forLp[0,1]hasbeenstudiedbyAbramovichandZaidenberg[1,2].TheyprovedthatifYisa(realorcomplex)rearrangement-invariantnonatomicfunctionspaceon[0,1]isometrictoLp[0,1]forsome1≤p∞thentheisometricisomorphismcanbeestablishedviaanidentitymap,i.e.kfkY=kfkpforallf∈Y(cf.also[5,3]forthecasewhenweadditionallyassumethatYisanOrliczspaceon[0,1]).Zaidenberg[16,17,14]studiedthegeneralformofisometriesbetweentwocomplexrearrangement-invariantspaces(r.i.spaces)andJamison,Kami´nskaandP.K.Lin[6]studiedisometriesbetweentwocomplexMusielak-OrliczspacesandbetweentworealNakanospaces.Theyprovedthatsur-jectiveisometriesinsuchsettingshavetobeweightedcompositionoperatorsandZaidenbergchar-acterizedsituationswhentheexistenceofisometrybetweencomplexr.i.spacesXandYimpliesthattheidentitymapisalsoanisometry.Theorem1belowgeneralizestheseresultstorealspaceson[0,1].WethenuseZaidenberg’scharacterizationofisometrygroupsofr.i.spacestocharacterizewhentheidentitymapbetweenrealr.i.spacesisanisometry.WealsogiveafulldescriptionoftheexceptionalcaseofOrliczspaceswhichcanbeisometricevenwhentheirOrliczfunctionsaredifferent(Corollary9providestherelationbetweentheOrliczfunctionsthathastobesatisfiedinthatcase).Acknowledgments.IwishtothankS.DilworthandH.Hudzikfordrawingmyattentiontothisproblem,S.Dilworthformanyinterestingdiscussions,Y.Abramovichforhisinterestinthiswork,andN.CarothersforhiswarmhospitalityduringmyvisitattheBowlingGreenStateUniversity,wherethisworkwasstarted.2.PreliminariesLetussupposethatΩisaPolishspaceandthatμisaσ-finiteBorelmeasureonΩ.WeusethetermK¨othespaceinthesenseof[12,p.28].ThusaK¨othefunctionspaceXon(Ω,μ)isaBanachspaceof(equivalenceclassesof)locallyintegrableBorelfunctionsfonΩsuchthat:(1)If|f|≤|g|a.e.andg∈Xthenf∈XwithkfkX≤kgkX.(2)IfAisaBorelsetoffinitemeasurethenχA∈X.TheK¨othedualofXisdenotedX′;thusX′istheK¨othespaceofallgsuchthatR|f||g|dμ∞foreveryf∈XequippedwiththenormkgkX′=supkfkX≤1Z|f||g|dμ.ThenX′canberegardedasaclosedsubspaceofthedualX∗ofX.Arearrangement-invariantfunctionspace(r.i.space)isaK¨othefunctionspaceon(Ω,μ)whichsatisfiestheconditions:(1)X′isanormingsubspaceofX∗.(2)Ifτ:Ω−→Ωisanymeasure-preservinginvertibleBorelautomorphismthenf∈Xifandonly2iff◦τ∈XandkfkX=kf◦τkX.(3)kχBkX=1ifμ(B)=1.Thecommonlystudiedr.i.spacesareclassicalLebesguespacesLp,Orlicz,LorentzandOrlicz-Lorentzspaces.Werecallthedefinitionsbelow.Wesaythatϕ:[0,∞)−→[0,∞)isanOrliczfunctionifϕisnon-decreasingandconvexwithϕ(0)=0.WedefinetheOrliczspaceLϕtobethespaceofthosemeasurablefunctionsfforwhichkfkϕisfinite,wherekfkϕdenotestheLuxemburgnormdefinedbykfkϕ=inf(c:ZΩϕ|f(ω)|cdμ(ω)≤1).Iffisameasurablefunction,wedefinethenon-increasingrearrangementofftobef∗(x)=supnt:μ(|f|≥t)≥xo.If1≤q∞,andifw:(0,∞)→(0,∞)isanon-increasingfunction,wedefinetheLorentzspaceLw,qtobethespaceofthosemeasurablefunctionsfforwhichkfkw,qisfinite,wherekfkw,qdenotestheLorentznormdefinedbykfkw,q=Z