DOC

Some

By Angela Wallace,2014-07-24 02:47
13 views 0
Some

    Some

App1.Math.J.ChineseUniv

    ;2008,23(1):4350

    ;Somenewintegralinequalitiesfor

    ;conjugateharmonictensors

    ;GAOHongya,HOULanru.,

    ;Abstract.Anewkindofweight.(1,2,Q)weightisusedtoprovethelocalandglobal

    ;integralinequalitiesforconjugateA-harmonictensors,whichcanberegardedasgeneralizations

    ;oftheclassicalresults.Someapplicationsoftheaboveresultstoquasiregularmappingsare ;given.

    ;?1Introduction

    ;ConjugateA-harmonictensorsareextensionsofconjugateharmonicfunctionsandp- ;harmonicfunctions,P>1.Inrecentyears,therehavebeenremarkableadvancesmadein ;thefieldofconjugateAharmonictensors.Manyinterestingresultsofthemandtheirappli

    ;cationsinfieldssuchaspotentialtheory,quasiregularmappingsandthetheoryofelasticity ;havebeenfound,see

    1-11].Formanypurposes,weneedtoknowtheintegrabilityofconjugate ;Aharmonictensorsandestimatetheintegralsforthem.Theaimofthispaperistoestablish ;twoweightintegralinequalitiesandtostudytheintegrabilityofconjugateAharmonic

    tensors

    ;andestimatetheintei2,?-.,2),1il<i2<…<il,f=0,1,…,n.TheGrassmanalgebra

    ;A=0…sagradedalgebrawithrespecttotheexteriorproducts.ForoL=?oLei?Aand

    ;=

    ;?ei?A,theinnerproductinAisgivenby(oL,)=?oLwithsummationoverall

    ;l-tuplesI=(i1,i2,…,il)andallintegersf=0,1,…,.TheHodgestaroperator:A__?A

    ;isdefinedforalloL,?Abytherule1=elAe2A…AenandoLA*flA,oL=(oL,)(1).

    ;ThenormofoL?AisgivenbytheformulaII2=(oL,oL)=(A)?A.=R.TheHodge

    ;Received:2005-0828.

    ;MRSub1ectClassification:31B05,58A10,46E35.

    ;Keywords:conjugateA-harmonictensor,A.(1,A2,~)-weight,weightedinequality,quasiregularmapping.

    ;DigitalOb{ectIdentifier(DOI1:10.1007/s1176600801063.

    ;SupportedbytheNaturalScienceFoundationofHebeiProvince(07M0031andtheDoctoralFundofHebei

    ;ProvincialCommissionofEducation(B20041031.

    ;

    ;App1.Math..ChineseUnivVo1.23,No.1

    ;starisanisometricisomorphismonAwith}:A__?Aand(1)():A__?A.Let

;1P<?.WedenotetheweightedLp—normofameasurablefunction,overEby

    ;,Ilp,E,.=(PwadxI,

    ;where/-formsonQisaSchwartzdistributionwithvaluesinA(R).ThesymbolD(Q,A) ;isusedtodenotethespaceofalldifferential/-forms.WewriteLP(12,A)forthe/-forms

    L(Q,R)forallordered ;()=?,wI(x)dxI=?o2i…tz(x)dxi1Adxi2A…Adxiwith0dI?

    ;/-tuplesI.ThusLP(12,A)isaBanachspacewithnorm

    ;Q=

    ;(圳如1/p=((?(p/21/p

    ;Similarly,W1,(Q,A)arethosedifferential/-formsonQwhosecoefficientsareinW1,(Q,R). ;Thenotations1

    ;.

    ;,

    ;c

    ;p(Q,R)and.1,cp(Q,A1)areself-explanatory.Wedenotetheexteriorderiva- ;tivebyd:D(Q,A)__‚D(Q,A+)forl=0,1,…,n.Itsformaladjointoperator

    ;d:D(Q,Al+)__?D(Q,A)isgivenbyd=(1)+donD(Q,AI+I),l=0,1,…,n.

    ;Therehasbeenremarkableworkinthestudyofthe.A-harmonicequation ;A(x,dw)=0

    ;whereA:Q×A()__?A(R)satisfiesthefollowingconditions:

    ;IA(x,)InIIand(A(x,),)(1.2)

    ;foralmosteveryx?Qand?A(R).Herea>0isaconstantand1<P<?isafixed

    ;exponentassociatedwith(1.1).Asolutionto(1.1)isanelementofSobolevspace.1,cp(Q,A)

    ;suchthat

    ;/(,),)=0

    ;forall?.1,

    ;c

    ;p(Q,A)withcompactsupport.

    ;Definition1.1.WecallanAharmonictensorinQifsatisfiestheharmonicequation ;(1.1)inQ.

    ;Adifferentialfform?D(Q,A)iscalledaclosedformifdu=0inQ.Similarly,a ;differential(1+1)formv?D(Q,A+)iscalledacoclosedformifdv=0.Adifferentialform ;iscalledap-harmonictensorif

    ;(Idolp.du1=0andd*u=0

    ;where1<P<?.Theequation

    ;(,du)=dv(1.3)

    ;iscalledtheconjugate.A-harmonicequation.Forexample,du=dvisananalogueofa ;CauchyRiemannsysteminR.Clearly,theAharmonicequationisnotaffectedbyaddinga

    ;closedformtoandcoclosedformtov.Therefore,anytypeofestimatesbetweenandv ;mustbethemoduloofsuchforms.Supposethatisasolutionto(1.1)inQ.Then,atleast ;locallyinaballB,thereexistsaform?WLq(B,AI+I),1+1=1,suchthat(1.3)holds.

    ;Definition1.2.Whenandvsatisfy(1.3)inQ,andAexistsinQ,wecallandvconjugate

    ;AharmonictensorsinQ.

;

    ;GAOHongya,eta1.SomenewintegralinequalitiesforconjugateAharmonictensors45

    ;Definition1.3.Wecallap-harmonicfunctionifsatisfiesthep-harmonicequation ;div(VulVulP,=0

    ;withP>1.Itsconjugateintheplaneisaq-harmonicfunction,1+1:1,whichsatisfies

    ;ll2=(Ov,Ov).

    ;NotethatifPq=2.wegettheusualconjugateharmonicfunctions. ;WewriteR=R.BallsaredenotedbyBandaBistheballwiththesamecenterasBand ;withdiam(aB1=6~diam(B).The礼一

    dimensionalLebesguemeasureofasetERisdenoted

    ;bylE1.Wecallwaweightifw?c(Rn)andw>0a.e.Alsoingenerald=wdxwherew ;isaweight.Wecanfindthefollowingresultin[8]:LetQcRbeacubeorabal1.Toeach

    ;Y?QthereiscorrespondinglyalinearoperatorKv:ca(o,,A)_.?(Q,A)definedby ;andthedecomposition=d()+().

    ;WedefineanotherlinearoperatorTQ:(Q,A)_.?(Q,A)byaveragingKvoverall ;pointsYinQ,TQ”,=J_.()Kvwdy,where?c(()isnormalizedbyJ_Q()d=1.We

    ;definethefformQ?D(Q,A)by

    /()d,ifl=0,andQ=d(),ifl=1,2,…,, ;Q=lQl

    ;dQ

    ;forall?LP(Q,A),1P<...

    ;?2Local.(1,2,)weightedintegralinequalities

    ;Definition2.1.Wesaytheweight(Wl(),w2())satisfiesthe.(1,2,Q)conditionfor

    ;somer>1andl,2,A3>0,write(Wl(),w2())?.(1,2,Q),ifWl()>0,w2()>0, ;a.eand

    ;,,,

    ;s

    ;()1()~2/(T-1)d)<?foranyballBCQ.

    ;Ifwechoosewl=w2=wand1=2=A3=1,wewillgettheArweight,see[12,13]for

    ;thebasicpropertiesofArweights.ChoosingWl=w2=w,l=and2=3=1,weget ;theAr(,Q)weightsintroducedin[2].ChoosingWl=w2=w,1=2=1andA3=,we ;obtainthe(Q)weightsintroducedin[14].See[12,13]forthebasicpropertiesof

    weights.

    ;Weneedthefollowinglemmainthispaper.

    ;Lemma2.1.Ifw?,thenthereexistconstants>1andC,independentofw,suchthat ;llBCIBI(z)/Zllwll1,B

    ;forallballsBCR.

    ;InthispaperwewillneedthefollowinggeneralizedHSlder’sinequality.

    ;Lemma2.2.Let0<<..,0<<..ands=+.Iffandgaremeasurable

    ;functionsonR,thenlI/gll,llfll,?lI9llforanyQCR. ;In[10],NolderobtainedthefollowinglocalCaccioppolitypeestimate.

    ;

    ;App1.Math.3.ChineseUnivVl0l_23.No.1

    ;TheoremA.LetbeanharmonictensorinS2ando->1.Thenthereexistsaconstant ;C,independentofanddu,suchthatIIdulls,

Cdiam(B)_1Iluells,crBforallballsorcubes

    ;Bwitho-BCQandallclosedformsc.Here1<s<?.

    ;ThefollowingweakreverseH51derinequalityappearsin[10]. ;TheoremB.LetbeanharmonictensorinQ,o->1and0<s,t<?.Thenthereexists

    ;aconstantC,independentof,suchthatllulls,BCIBI(t-s)/Ilull,,Bforallballsorcubes ;Bwitho-BCQ.

    . ;Thefollowingresultappearsin[5

    ;TheoremC.ForarbitraryP>0,thereexistsaconstantC,suchthat ;rf/Iu(O)lPdxdyC/Ivu(0)Idxdy

    ;jDjD

    ;forallanalyticfunctionsf=+ivinthediskD.

    ;Nolder[4generalizedtheabovetheoremandprovedthefollowingimportantresult. ;TheoremD.LetandvbeconjugateharmonictensorsinQCR,>1and0<s,t<?.

    ;ThenthereexistsaconstantC,independentofandv,suchthatforarbitraryballorcubeB, ;BCQ.wehave

    ;IluUBII,BCIBIZlIvc1q.

    ;/p

    ~p/qIluc2lI:, ;BandIIvvBII,,BGIRl--

    ;where,=1/s+1In(1/t+1/~)(q/p),c1isanydifferentialformin(Q,A)andc1=0, ;c2isanydifferentialformino1,

    ;cq(Q,A)anddc2=0.

    ;Fromtheabovetheoremswecangetthefollowingresulteasily. ;TheoremE.LetandvbeconjugateharmonictensorsinQCR,P>1.Assumethat ;1<s<?isafixedexponentassociatedwiththeharmonicequation,1<t<?.Then

    ;thereexistsaconstantC,independentof,vanddu,suchthatforallballsorcubesBwith ;pBCQ,wehave

    ;1Idull,BClBIc,

    ;where,=1/s(1/t+1/n)(q/P),cisanydifferentialformin(Q,A)anddc=0. ;WenowgeneralizeTheoremEtothefollowinglocalweightedestimate. ;Theorem2.1.LetandvbeconjugateharmonictensorsinQCRandP>1.Assume ;that1<s<?isafixedexponentassociatedwiththeharmonicequation,1<t<?and

    ;(1,w2)?A.(1,2,Q),叫?A(Q)forsomer>1andhi,A2,3>0.Thenthereexists ;aconstantC,independentof,vanddu,suchthatforallballsorcubesBwithpBCQ,we ;have

    ;where,=1/s

    ;Notethat(2.1)canbewrittenas

    ;(.-ds叫如l/scB(B-c-tA2A3/qSdx).

    ;Proof.Since叫?,forsomer>1,byLemma2.1thereexistconstantsOL ;suchthat

    ;lIll.,BCIBI(.)/.lIII,,

    ;B

    ;(2.1)

    ;dc=0.

;(2.1)

    ;>1andC1>0 ;(2.2)

    ;m

    ;?

    ;c

    ;,m

    ;m

    ;

    ;.

    ;d

    ;{j

    ;C

    ;Vl

    ;{i

    ;

    ;GAOHongya,eta1.SomenewintegralinequalitiesforconjugateAharmonictensors

    ;foranycubeorballBcR”.Because1/,=l/as+f1)~as,byLemma2.2wehave

    ;liduII,

    ;<ll…1/s..duII/(Q—1) ;Next,choosem=qst/(qs+ptk3(r1)),thenm<t,byTheoremEwehave

    ;Ildull/(1),BlBlIIuc[[q/ ;,

    ;p

    ;B,

    ;wherel=f1)~as(1/m+1/n)(q/p).Combining(2.2),

    ;Idull,

    ;B,

    ;C~IBIllll1/,Bsll”

    ;(2.4),wehave ;47

    ;(2.3)

    ;(2.4)

    ;(2.5)

    ;ll”cllB=(

    ;B

    ;l”cld)Pm=(

    ;B

    ;(1”clP”k2”ka/q8”WA2)d)pm ;(

    ;B

    ;c1t;tA2Aa/qSdx( ;B

    ;((qs)mt/(t-m)dxm)

;=IIv-cllq/p.’

    ;(

    ;B

    ;()r_)卜,?

    ;(2.6)

    ;Combining(2.5)and(2.6)wehave ;Ildull,

    ;B,

    ;C~IBI2?IIv-cllq/p(l/s ;

    ;(

    ;B

    ;(1

    ;

    ;~2/(r_Udx)

    ;Since(Wl,W2)?.(l,2,2),wethenhave ;(‘(

    ;B

    ;(r__

    ;(

    ;B

    ;)s’(

    ;B

    ;()

    ;_((/’(B(/(r-1)dxJpD~p

    ;)qs

    ;_(1Hqs(m)/(

    ;B

    ;(

    ;B

    ;(/s

    ;(2.7)

    ;(2.8)

    ;Substituting(2.8)into(2.7),wefindthIldull,B, ;CIBIIIv-e[[q/v

    ;B,

    ;forallballs

    ;orcubesBwithpBC2,=1/s(1/(+1/n)(q/p)andanydifferentialformcin1.,cp(2,A)

    ;withdC=0.ThisendstheproofofTheorem2.1.

    ;Notethattheparametersl,2andA3inTheorem2.1areanyrealnumberswithl,2,A3>

    ;0.Thereforewewillhavedifferentversionsoftheweightedestimatesbychoosingl,2and

    ;3tobedifferentvalues. ;Ifwechoosel=1inTheorem2.1,wewillhavethefollowingresult.

    ;Corollary2.1.LetUandVbeCOnjugateAharmonictensorsin

2CR”andP>1.Assume

    ;that1<8<?iSafixedexponentassociatedwiththeharmonicequation,1<t<?an

    d

    ;(Wl,W2)?.(2,2),Wl?(2)forsomer>1and2,A3>0.Thenthereexistsaconstant ;C,independentofU,Vanddu,suchthatforallballsorcubesBwithpBcwehave ;dull,B,CIBlZllvclIq/Plt,pB,2p^./(2.9)

    ;

    ;App1.Math.J.ChineseUnivVl01.23.No.1

    ;where:1/s(1/(+1/n)(q/p)andcisanydifferentialformin.1,cp(,A)withd}c=0. ;Similarly,wecangettheresultswhen2=1or3=1,soitwillnotbementionedhere. ;Ifwechoosesomespecialnumbers,wewillgetthefollowingsymmetricversions.Forexample,

    ;choosinghi=1and2=qs/pth3inTheorem2.1,wehavethefollowingresult. ;Corollary2.2.LetandvbeconjugateharmonictensorsincRandP>1.Assume ;thatl<s<o.isafixedexponentassociatedwiththe

    harmonicequation,1<t<o.and

    ;(Wl,w2)?.(hi,2,),叫?Al(a)forsome7’>1,1=1,h2=qs/pt,~3andsome3>0.

    ,vanddu,suchthatforallballsorcubesB ;ThenthereexistsaconstantC,independentof

    ;withpBC.wehave

    ;fIdYll,.BClBI~cll,(2.1O)

    ;where:1/s(1/(+1/n)(q/p)andcisanydifferentialformin(,A)withd}c=0. ;Choosinghi=1/pt,2=1/pand3=1/t,weobtainthefollowingresult. ;Corollary2.3.LetandvbeconjugateAharmonictensorsincRandP>1.Assume ;that1<s<o.isafixedexponentassociatedwiththe

    harmonicequation,1<t<o.and

    ;(w1,w2)?.(hi,2,),叫?Al(a)forsome7’>1,hi=1/pt,2=1/pand3=1/t.

    ;ThenthereexistsaconstantC,independentof,vanddu,suchthatforallballsorcubesB ;withpBC.wehave

    ;1/tClBIpcl

    ;,

    ;(2?11)

    ;where=1/s(1/(+1/n)(q/p)andcisanydifferentialformin.1,.p(,A)withd}c=0. ;Ifwechoosehi=qs,h2=qand3=s,wethenhavethefollowingresult. ;Corollary2.4.LetandvbeconjugateharmonictensorsincRandP>1.Assume ;that1<s<..isafixedexponentassociatedwiththeA-harmonicequation,1<t<..a

    nd

    ;(Wl,w2)?.(hi,2,),叫?Al(a)forsome7’>1,hi=qs,h2=qand3=s.Then

    ;thereexistsaconstantC,independentof,vanddu,suchthatforallballsorcubesBwith ;pBC.wehave

    ;}sCIBIpcllt,(2.12)

    ;where:1/s(1/(+1/n)(q/p)andcisanydifferentialformin1.,.p(,A)withd}c=0. ;?3Globalweightedinequalities

    ;WeneedthefollowingpropertiesofWhitneycoversappearingin[10]toprovetheglobal ;results.

    ;Lemma3.1.EachQhasamodifiedWhitneycoverofcubes=Qtsuchthat ;UQ,?)(Q?)(Q

    ;iQ?

    ;forallx?RandsomeN>1,andifQnQj??,thenthereexistsacubeR(thiscubedoes ;notneedtobeamemberof)inQinQjsuchthatQtUQjcNR.Moreover,ifisa-John,. ;thenthereisadistinguishedcubeQ0?whichcanbeconnectedwitheverycubeQ?bya

    ;chainofcubesQ0,Q1,…,Q=QfromandsuchthatQcQt,i=0,1,2,…,k,forsome

    ,). ;o-=(

    ;

    ;GA0Hongya,eta1.SomenewintegralinequalitiesforCOnjugateA-harmonictensors49

    ;WenOWprovethefollowing.(l,2,Q)weightedintegralinequalityinaboundeddomain ;Q.

    ;Theorem3.1.LetandVbeCOnjugateM-harmonictensorsinQCRandP>1.Assume ;thatl<s<..isafixedexponentassociatedwiththe4

    harmonicequation.1<t<..and

    ;(l,W2)?.(l,2,Q),?Al(f1)forsomer>landl,2,3>0Thenthereexistsa ;constantC,independentof,Vanddu,suchthat

    ;d

    ;,

    ;Clal[V--Cnyakinaseriesofarticles

    ;thatbegantoappearin1966.Seealsothemonograph[16]fordetails.Quasiregularmapping

    s

    ;areinterestingbecauseofnotonlytheresultsobtainedaboutthem,butalsomanynewideas

    ;generatedinthedevelopmentcourseoftheirtheory.Itisknownthatiff(x)=(f,f,…,t”)

    ;isK—quasiregularinR”,then

    ;=fldfAdfA…Adfand=,fl+ldf+A…Adf”.

    ;wheref=l,2,…,n,areconjugateA—harmonictensorswithP=n/zandq=n/(nf),andA

    ;issomeoperatorsatisfying(1.2).

    ;Let(Wl,W2)?.(l,A2,Q)and?A(Q),byTheorem2.1weobtainthefollowing ;localweightedintegralinequalityforquasiregularmappings: ;(/ld(t厂八八八一)ld)/I,B

    ;r

    ;CIBI~(/l:1cfl+ldf+-??cl../dx)q/P,JDB

    ;whereCisindependentoff,=1/8(1/t+1/n)(q/p)anddC=0.

    ;Similarly,byTheorem3.1weobtainthefollowingglobalweightedintegralinequalityfor

    ;quasiregularmappings:

    ;AdfA…Adf)ldx)/

    ;whereC,andCareasabove

    ;:1cfl+ldfA…Adf”cJ’tA2A3/dx)q/p,

    ;0

    ;

    ;50App1.Math..ChineseUniv.Vl01.23.No.1

    ;References

    ;1BallJM.Convexityconditionsandexistencetheoremsinnonlinearelasticity.ArchRatio

nal

    ;MechAnal,1977,63:337403.

    ;2BallJM,MuratF.W

    quasiconvexityandvariationalproblemsformultipleintegrals.JFunct ;Anal,1984,58:225253.

    ;3DingS.WeightedHardyLittlewoodinequalityforAharmonictensors,ProcAmerM

    athSoc,

    1735. ;1997,125:1727

    ;41waniecT,MartinG.GeometricFunctionTheoryandNonlinearAnalysis,In:OxfordMath

    ;Monographs,NewYork:OxfordUnivPress,2001.

    ;5NolderCA.AquasiregularanalogueoftheoremofHardyandLittlewood,TransAmerMath

    ;Soc,1992,331(1):215226.

    ;6StroffoliniB.Onweaklyharmonictensors,StudiaMath,1995,114(3):289301.

    ;71waniecT.p-harmonictensorsandquasiregularmappings,AnnMath,1992,136:5896

    24.

    ;81waniecT,LutoborskiA.IntegralestimatesfornullLagrangians,ArchRationalMechAnal,

    ;1993,125:2579.

    ;91waniecT,MartinG.Quasiregularmappingsinevendimensions,ActaMath,1993,170:2981.

    ;10NolderCA.HardyLittlewoodtheoremsforAharmonictensors,IllinoisJMath,199

    9,43:613

    ;631.

    ;11NolderCA.Acharacterizationofcertainmeasuresusingquasiconformalmappings,ProcAmer

    ;MathSoc,1990,109(2):349456.

    ;12GarnettJB.BoundedAnalyticFunctions,NewYork:AcademicPress,1970. ;13HeinonenJ,KilpelainenT,MartioO.Nonlinearpotentialtheoryofdegenerateellipticequations,

    ;Oxford:ClarendonPress,1993.

    ;14DingS.Newweightedintegralinequalitiesfordifferentialformsinsomedomains,PacificJMath,

    ;2000,194(1):4356.

    ;15HardyGH,LittlewoodJE.SomepropertiesofCOnj.ugatefunctions,JReineAngewMath,1932,

    ;167:405.423.

    ;16NolderCA.GlobalintegrabilitytheoremsforAharmonictensors,JMathAnalAppl,2

    000,247:

    ;236-245.

    ;17DingS.ParametricweightedintegralinequalitiesforAharmonictensors,Zeitschriftf

    urAnalysis

    ;undihreAnwendungen(JournalofAnalysisanditsApplications),2001,20:691708.

;18StaplesSG.LaveragingdomainsandthePoincardinequality,AnnAcadSciFennSerAI

    Math,

    ;1989,14:103127.

    ;19ShiP,DingS.TwoweightHardyLittlewoodinequalitiesandCaccioppolitypein

    equalitiesf?

Report this document

For any questions or suggestions please email
cust-service@docsford.com