局部域上的Herz型Besov空间

局部域上的Herz型Besov空间 第20卷第2期 2003年6月 经济数学 MATHEMATICSINECONOMICS Vo1.20No.2 June.2003 HERZ—TYPEBESOVSPACESoN LoCALFIELDS. TangCanqin (Department.ofMathematics,HunanUniversity,Changsha,Hunan,P.R.Chi na; DepartmentofMathematics,ChangdeNormalUniversity,Changde,Hunan, P.R.China) LiQingguoMaBolin (DepartmentofMathematics,HunanUniversity,Changsha,Hunan,P.R.Chi na) AbstractLetKbealocalfieldandKbethe—dimensionalvectorspaceoverK.Inthispaper.theauthor discusssomepropertiesofK:?B(). KeywordsHerz-typeBesovspace,localfield.? ?1.Introduction BesovspaceOnRplaysanimportantroleinthestudyoffunctionspaces.Forthe classicalcaseonecanseeE4-1,E23andSOon.Peoplealsopayattentiontothiskindofspace onlocallyVilenkingroupsorlocalfields.TheproperitiesofBesovspacesonlocallycompact Vilenkingroupsorlocalfieldsweregivenin[3]and[6].RecentlyXu(Es])introducedthe Herz—typeBesovspacesonRandgiveaunifiedapproachforHerz—typeBes ovspaces. Motiviatiedbytheirwork,weconsidertheproperitiesofHerz—typeSpaceon localfields. Throughoutthispaper.KwilldenotealocalfieldandKbethe—dimensionalv ector spaceoverK.TheabsolutevalueJJofz?Kisasusual,andforz一(zl,z2,…,) ?K,z. ?K,i一1,2,…,,whereIzI—max.{I五 I},dxistheHaarmeasureonKsuchthatl0l一1 andfBIf—,whereO一{37?K:fzf?1},BI一{z?K:ff?}and一 Pforsome primeintegerPandC?N.S()andS()aretestfunctionclassanditsdistributionspace respectively.Lp()and.qisasusual(see[6]).LetFfandFfbetheFourierand inverseFouriertransformof厂? S().Foranonnegativeintegera,the”a—orderpseudo— derivative”off?S()iSdefinedby D.f—F(<Y)F厂) inthedistributionsense,here<>一max{1,IzI}.the”a—orderpartialpseudo—derivative” aboutXiiSdefinedby ?Received:2002一l0—06 第2期汤灿琴李庆国马柏林局部上的Herz型Besov空间一63一 „丢厂一((. Inthispaper,theauthorwilldiscusstheproperitiesoftheHerz—typeBesovspaces :?B().Infact,when口一0,q—P,qa?B()一B(),thenitturnedtobethecase whichdiscussedin[6]. ?2.MainResultsandProof Wefirstrecallsomedefinitions. Definition1([6])Let()bethesetofallsystems{伤(z)}0cS()suchthat (i) (ii) ?(z)一1;J=o %c*z:z?K,lzl?}, 饬c{z:z?K”,.?lzl?}, (iii)foreverynonnegativeinteger口,thereexistsapositivenumberCnsuch 仍(7,.)l口ll?C.forJ一1,2,…, wheretheSobelevspace()aredefinedas ()l{厂:f?Lp(),IIfIIuc一?IIDII<o..)0?口?m ZhouandSugivethedefinitionofBesovspaceonlocalficldsasfollowsin[6] B. ()一{f:f?S(), IIfII. 一 IIF_.~Ffll()II一(?II3i,F一F厂II<o.) where伤一一卜1for一1,2,…,and一 0.The{}isthecharacteristicfunctionofB inK.Onthebasisofthis,wecangivethedefinitionofHerz—typeBesovspace. Definition2Let?R,O<p,q,?o.,口+1/q~O.Then :?B;()={厂?S():IIfIIx:?:一IIF一纺F厂l(Ka?)II<o.), withtheusualmodificationif一o.. Inthissection,wewillprovethatthequasi—normsIIfIIxisindependentonthe choiceofthesystem?(“). Theorem1Let一{访}0?(“),一{}0?().LetsERand0<8,P,g?o.,口 +1/q>0, (i)IIfII?andIIfII%?reequivalentquasi—normsonK:Bs? (ii)SpaceB;()isindependentofthechoiceof(). Proof.(i)ItiseasytoseethatbothIIfII?andIIfII,qBarequasi—norms?As (4)in[6], 64经济数学第2O卷 l F-?Ff:?F-?FF一1Ff, replacing/,and矿in(9)in[63byF一+,,andrespectively?w.bm . I1F一FF一-+,Ffx,I1?cI1(17一J.)IluIIF-l~j+rFfZrII ?CI1F,+,F/fl1 here77satisfiedthat:iffeLp(B),thenfO]一?)ELp(o)?andhn? l1F一-F一-Ff一{?.(一IlF-?%FFj+~FfZ,II) ?C{?.(llF-?,FfxfII)lip = CI1F+,Fflly, .rJ一?一andr-~---1?.?? 1lI1.,atedfr.mb.,,ebyII/II惫..Thfthisprovedthatfcanbeestimereore,l1I1edtrOmaDOv.y”%. Hense,qiequivalenttoeachother.uchthatnd一一fOr产 y三?2andsuchthat1,,…,={}o?()一o()nd一?… „2?一 e here 一 k ? > . O . . Fc:F一,+一.,similart.thepr..f.fProposSiFF(FCjFfXe)imiti.n2.1.2,nce一 +胁c:F一+,一?oheprooo?儿?J儿上? IIF-1~gj+rFFCjFfZ,?Cl1_.offxfllf.rO?r?愚 IIF-?q~jFf一ll-1.F,~FfZ,1l(L)II ?Cll3-1.F一FfZ,16(L)ll furthermore, 0ntheotherhand, fl1.,一llF,%Ffll(Kat)Il ?CllF~,jFfllp(K~) 一 ClIfIl. F-?Ff—F一+Ff,forJ>O and F-?F厂一?F一.够F厂, Similartoaboreproof,wecanshowthat l1F-1~,jFF一-+FfxfIl?CllF-?%FfZflI, thusl1fIl殳:.anbeestimatedbyClIfll莨:,一?Therf.re(ii)ip.ed? InthefOl1owing,A1CA2alwaysmeansthatthetopologilpA1 embedde ; dinthetopologicalspa . ceAz heembeddin?ties.fHerzspace(se Uthesingproposition2.2.1in[63andthmbedmgproperlues”,…… 口 U O U n tn O C 第2期汤灿琴李庆国马柏林局部上的Herz型Besov空间一65一 E13),theorem2iseasytobeproved.Hereweomittheproof. 1 Theorem2Let,oo<s<oo,0<p,g<o.and口>一寺, (i)If0<81?2?o.,thenKa邱,()cKB,(); (ii)If0<81,2?o.,e>o,thenKB?()c一Bs,(); (iii)If口1<口2,thenKB?a(K)EKe,?邱(K); (iv)If6?f,thenKB?()cKB?(); ,1 (V)Ifql?g2,thenKo:~B?()CKB?(),wherer一口一l去 Inthenextpart,wewilldiscusstheliftingpropertyofK.a? number,definetheopeatorIas .f=F一((>Ff),f?S(). Itiswell—knownthatisaone—to—onemappingfromS() 一 B(K).If盯isareal ontoitself.Itiseasyto seethatII一I+. Theorem3LetS,,口,P,qasinTheorem1.Suppose盯?R,m一 1,2,3,…,wehave (i)mapsK:B;()isomorphicallyontoK:BT()andII—II?一(r)is equivalentquasi—normonKqaB(). (ii)?.?《IIV?fIImandIIfIIm十?::areequvalient quasi—normonK:B(). Proof(i)Take一{}o?()suchthat一oand一J一r1for一1,2,….If fEK:?B?a(K),then IIjII?=II?一F一(?YFfII~(K~)I1 where =(?.II.一4F,(?>F/)了1, ?ooII~jO-a)F一(?>VfIIn一?.(?.II一F一(?>Ffz Therefore .. ..:..一 .. 厶i: (?.『I一F-?9;FfZt『I); IIF%FfII 户 IILfIIy—IIfII. ThisimprovesthatmapsK:?B()intoKqaB7(). Morever,ifgEK:By(“),thenf=i一g?K:B()and IIfII一II一gII一IIgII一IIII? Becauseisaone—to—onemappinginS(),thenitisalsoaone—to—onemappingfrom mapsKqa?B(“)ontoK:B7(“),andIIII(Kn)isequivalentquasinormon K:B?(“). 一 66一经济数学第2O卷 (ii)SinceB(K)cKB?AK)forany(>O,wehave llDfllKs,一m=llf?一?Cllfll?一 foranyO?y?m.Therefore II厂IIx??IID厂II,一+y?CII厂II?一?O??m ItiStrivalthatIIfII+??(n+1)IIfII?Bythe properitiesofquasi—norms,wealsoobtain IIfII叶?l SotheproofofTheorem3isfinished. [1] [2] [3] [4] [s] [6] References l?::.n lI.fllx lf?一. 口 BarernsteinII.A.andE.T.Sawyer,EmbeddingandmutlipliertheoremsforH()?MemoirsAmer. Math.Soc..59,Amer.Math.Soc.,ProvidenceR.I..1985. Bui,H.Q..RepresentationtheoremandatomicdecompositionofBesovspaces. Onnewer.C.W.andW.Y.Su,Homogeneous,BesovspacesonlocallycompactVilenkingroups, StudiaMath.T.,XCIII(1989).17,39. Frazier,M.andB.Jawerth,DecompositionofBesovspaces,IndianaUniv.Math.J.,34(1985),777 —— 799. Xu,J.S.HerztypeBesovandTrieber—Lizorkinspaces.preprint. Zhou,G.C.andW.Y.Su,ElementaryaspectsofB;.口()andF.口 ()spaces,Approx.Theoryand itsApp1.,8(1992),l1,27. 局部域上的Herz型Besov空间 汤灿琴李庆国马柏林 (湖南大学数学与计量经济学院,长沙,410081) 摘要令为局部域,K为K上n维向量空间.本文讨论Herz型Besov空 问K(K)的某些性质? 关键词Herz型Besov空间,局部域 CCC ??一 m p ?口 K