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SOLVABILITY IN HARDY SPACE FOR SECOND ORDER ELLIPTIC EQUATIONS

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SOLVABILITY IN HARDY SPACE FOR SECOND ORDER ELLIPTIC EQUATIONSIN,in,HARDY,SPACE,FOR,ORDER,Hardy,space,for,order

SOLVABILITY IN HARDY SPACE FOR

    SECOND ORDER ELLIPTIC

    EQUATIONS

    Ana1.TheoryApp1.

    Vo1.25,No.4(2009),355-368

    DOI10.1007/sl0496-009-0355X

    SOLVABILITYINHARDYSPACEFORSECONDORDER

    ELLIPTICEQUATIONS

    YongzhongSunandWeiyiSu

    (NanjingUniversity,China

    ReceivedJuly1,2009

    Abstract.WeobtainaprioriestimatesandsolvabilityinHardytypespaceinabounded domainOfRforsecondorderellipticequationswithcoefficientsoflimitedsmoothness? SucharesultcanbeservedasanendpointcaseoftheclassicalLP(1<P<oo)theoryfor secOndOrderellipticequations.Ourapproachisbasedonastandardtechniqueofperturba. tionratherthanthatofintegralrepresentationformula.

    KevWOrds:Hardyspaceinboundeddomain,secondorderellipticequation,?priori

    estimate,solvability

    AMSf2000)subjectclassification:35J15,35J25,42B30

    1IntroductionandStatementoftheMainResult

    Thepresentpaperisasubsequentworkofourearlierone([14]).Wecontinuetodiscua priorestimateandsolvabilityinHardytypespaceforlinearsecondorderellipticequationswith

    coefficientsoflimitedsmoothnessinaboundeddomain.Considerthefollowingsecondorder linearellipticoperator

    ,l

Ebi(x)Oiu+C(X)U.?Q

    f=1

    Here~CRnisandeddomain,Diu=,%=

    measurablefunctionssatisfying,fori,J=

    aij(x)=aft(x),

    2

    OXiOXj,andaij,bi,Carerealvalued

    ,

    n,thereexistconstants0suchthat

    II,V考?Rn,a.e.?Q;]aijI,Ibil,tcl.'(1.2)

    SupposedbyNNSFofChinaGrantNo.10571084andbyNNSFofChinaGrantNo.10771097

    .

    +

    .JD

    .J

    ?户

    =

    L

    .f

    ,,I\

    .J

    

    ?

    356ZSunetalSolvabilityinHardySpaceforSecondOrderEllipticEquations

    BasedontheclassicalLptheoryofCalder6nandZygmund([2],3)forsingularintegralsand Korn'stechniqueofperturbation,onecanshowthatifaij?c()andQisaCdomain,then thefollowingaprioriestimateholdfor?1'p()Nw,p(Q).

    D"II()C(1ltull(Q)+Il"lI(Q)),1<P<oo,(.3) withtheconstantCdependsonlyon,P,,Qandthemoduliofcontinuityofthecoefficients

    aij.ThesolvabilityofDirichletproblemforLinSobolevspacesW2,p(Q)isthenaconsequenc

e

    ofsuchanestimate.See,forexample,Theorem9.11,9.13and9.15in10].

    Itiswellknownthatthecorrespondingestimate(1.3)doesnotholdwhenP=1,evenfor theLaplacianoperatorL=A.Developmentsinharmonicanalysissuggestthatonemayreplace

    (Q)bysomesuitableHardytypespace.Infact,thetheoryofHardyspacesinadomain andcorrespondingestimatesfortheLaplacianoperatorwerewelldevelopedin4,5.Forthe

    generaloperatorLwithbi,c=0,i=1,2,,,z,theauthorsin6]provedanh1-typeestimate

    whenaijareDinicontinuous.

    InthepresentpaperweobtainaprioriestimateforsecondderivativesofasolutionuinHardy space^(Q)undertheassumptionthatu?c0n()andbi,c?cn()(seethenextsectionfor

    theexactdefinitionofthesefunctionspaces).Moreprecisely,wehave Theorem1.1.SupposeQisboundedC3-domaininRandLsatisfies(1.2lfaij?c(Q)

    andbi,c?Cln(),thenfor?01,(Q)n2,(Q)

    D2ull^,()~C(1ltQ/+liDQ)+^t(Q)),(1.4)

    whereCisaconstantdependingonn,J;L,

    ,Qandthemoduliofcontinuityofaij,bi,cinCln(Q).

    AsaconsequenceweobtainthefollowingresultaboutthesolvabilityofDirichletproblem forLinHardyspace.

    TheoI1em1.2.LetLandQbeasinTheoremJ.Jwithc(x)0.Iff?h1(Q),g?h,(Q),

    thentheDirichletproblemLu=,"g?t'(Q)hasuniquesolution?h2,1(Q).

    SeethenextsectionforthedefinitionofHardyandHardySobolevtypespaces.

    Since()containsDinicontinuousfunctionsasapropersubclass,ourresultsimproved thosein6.Infact,theauthorsin[6]suggestedthataijbelongingtothespaceofvanishing LMOmaybeenoughtogettheestimate.Notethat(Q)isapointwiseversionandasubspace ofvanishingLMO.SuchaconjectureisbasedontheworkofF.Chiarenza.M.FrascaandE Longo([7,8),whereitisshownthat(1.3)arestilltrueifaij?VMO,thefunctionspaceof

    vanishingmeanoscillationfirstintroducedbyD.Sarasonin13],whichcontainsc(a)asa

    Ana1.TheoryApp1.,Vo1.25,No.4(2009)357

propersubspace.Alsointhepreviouswork[14,theinteriorh1-estimateisobtainedunder

    moregeneralassumptionthataijbelongtothespaceofvanishingLMO.Unfortunately,wecan notobtainboundaryestimatesunderthesameassumptionbecauseofthelackofcorresponding

    estimatesfortheboundaryintegraloperators.Formoredetailsonecanreferto6,8]and

    14].

    Anewcontributionofthepresentpaperisaboundaryestimateunderourassumptionon coefficientsofL.Inordertogettheboundaryestimate,weproceedalongthelineofLp(1< P<oo)estimatesasshowninChapter9,[1O].Unlikeearlierworkssuchas[4,5,6and[8,

    whereaprioriestimatesareobtainedviaexplicitintegralrepresentationformula,ourapproach

    isbasedonstandardtechniquesdevelopedinthetheoryoflinearellipticequations.Wefirstget aprioriestimateforLwithconstantcoefficientsbasedoncorrespondingestimateforLaplacian,

    andthenuseperturbationtechniquetohandleLwithvariablecoefficients. Thepaperisorganizedasfollows:wefirstintroducesomedefinitionsandpreliminaryresults inthenextsection.TheestimateforLaplacianandLwithconstantcoefficientsisprovedin Section3.FinallyinSection4wegivetheproofofTheorem1.1and1.2.

    2DefinitionsandSomePreliminaryResults

    TherealvariabletheoryofHardyspacesbeginswiththepioneeringworkofC.Fefferman andE.M.Stein([9]),wheretherealHardyspacesHp()(0<p1)areintroducedandstudied. ThelocalversionhP(R)(0<p1),whichismoresuitableforthestudyofellipticoperators, wasfirstintroducedbyD.Goldbergin[11.Onecanreferto[15formoredetailsaboutHardy

    spacesinR.ForHardyspaceshP(~-2)inadomain,see[4,5]and12].Inthefollowing

    webrieflyrecallthedefinitionsandpropertiesofh(R)anditsrestrictionh(Q)toadomain QCR.

    Let?cbeamollifier,whichisradialandsatisfysuppq~cB1(0),theunitballinR, 0and/:1.Definethemaximalfunction(1ocalmaximalfunction)offasf(x)= JRn

    1'.

    supI,)I()=sup.I,)1),where)=(}).TheHardyspaceH(R)O<f<ooO<t<1f'f (1ocalHardyspaceh(R))isdefinedasasubspaceofL(R)suchthatI厂?L(R)f?

    L1()),equippedwiththenornlII,llHl(n)=lI刘,?()(1lf[1h(Rn)=lIm~IIIL(R)).Fora

    domainQCRdefineh(Q)(denotedbyh(Q)in[5])astherestrictiontoQofafunction F?h(R)withthequotientnornl,i.e.,

    (Q)={f?L1(Q)JthereexistsF?hI(")suchthatFJQ=),

    withthenorlTlIlfllh(Q)=inf{IIFI}^IF?h1(),FJQ=}.

    358Z.SunetalSolvabilityinHardySpaceforSecondOrderEllipticEqua6ons SimilartoSobolevspaces,wedefinethecorrespondingHardySobolevspaces

    ,

    (Q)={M?W,(Q)ID?h(Q),0k),

    with}lullh-,'(Q)=?0,,lIIZ~ullh(Q).HereDistheweakderivativesoforderkandWm,1(Q) istheSobolevspaceofordermwithLl_integrability.

    Alsodefine0m'(Q)astheclosureof

    (Q)inh(Q).

    ThemostimportantpropertyofHardyspaceisitscharacterizationbyatomicdecomposition. Werecallthefollowing

    Definition2.1.LetQbeaboundeddomain.Letabealocallyintegrablefunctionwith compactsupportcontainedinaballBrCQ.IfahasitssupportingballBrwithr<1,B4rCQ andsatisfiesbothofthefollowingtwoconditions

    (i)I()lIB』一=Cnr,fora.e.?~2;(sizecondition),

    (ii)/a(x)dx=0;(cancellationcondition),

    thenaiscalledatype(a)atom.Afunctionaiscalledatype(b)atomifeitheritonlysatisfies thesizecondition(i)withthesupportingballBrsuchthatr<l,B2rCQ,B4,naQ?0orthe

    radiusofitssupportingballisnotlessthanone.Sometimewecallthefirstcaseoftype(b)atom asaboundaryatom.

    OneCanexaminethatbothtwokindsofatomsbelongtoh(Q)withnO1Tfllessthanorequal toone.

    Theorem2.1.[5112fEh(Q)ifandonlyifthere",

    +~o

l,{)el,type(a)atoms

    {nJ)andtype(b)atoms{)suchthat

    NN

    ,?liraV~'jaj+

    wherethelimitistakeninh(Q).Moreove~

    lI|Il-(Q)?inf(ElZjl+?)

    withtheinfimumbetakenoverallpossibleatomicdecompositionsoff. (2.1)

    Remark2.1.Thereisasimilaratomicdecomposition(withoutboundaryatoms)forafunc

    tionf?h(R).Moreover,ifwedefine

    JIz()={f?hI()Isuppfc)(2.2)

    with()=Ilfllh(),thenfortheatomicdecompositionoffEJIz(),thesupportingballs

    ofallatomscanbechosentObeincludedinQ.See4formoredetails.

    扣?

    +

    ?

    =

    Ana1.TheoryApp1.,Vo1.25,No.4(2009)359

    Thespaces6"1n()andc0n()aresubspaceofc()definedinthefollowingway.Denote

    by()thespaceofboundedcontinuousfunctiononRn.Let ):{f?c(R)lIJfllc,()<c.),

    fllc,()=lIflIL~IRn)+supIInIyllIf(x)一厂(y)I<oo

    lX--yl<l

    (2.3)

    ()==={fECln()[limsup,lilY()},(2.4)

    whereBrisanyballinRwithradius,.>0.ForadomainQcR,wedefineCln()(respectively

    c())astherestrictionofGn()(respectivelyc0n())to.

    (Q).Ontheother Remark2.2.ItiseasytoshowthatiffisDinicontinuousthenf?

    side,thereexistsf?()butfisnotDinicontinuous.Forexample,f(t)=sgn(t)[(1+

Ilnt[)(1+ln(1+IlnfI))]and=【一1,1].SeeProposition1.10in[1.

    Thefollowingpropositioniscrucialinourprocess. Proposition2.1.,,?Gn(),f?h(R),thenu/f?h(R)with

    ~fllh-(Rn)cI1wllc,(Rn)lIflib-(n),

    whereCisapositiveconstantdependingonlyonn.SimilarresultholdsifonereplaceRbyQ

    .

    e.,矿?(=l(),f?h(Q),then?h1(Q)with

    ~fllh-(Q)cll~llc,()IIfllh-(Q),

    TheproofofthispropositionisalmostwordforwordrepeatofProposition2.1in[14except

    forthemodificationofreplacingbylfr().Weomitthedetails. ThenextlemmaisessentiallyLemma5.1in14],butthereisamistakeintheproof.We takethisopportunitytogiveacorrectone.

    Lemma2.1.LetQbeaboundeddomaininR.Iff?h(Rn)withsuppfCCQccQ,

    then

    IIflib?(Q)IIflib()c(1+Ilndist(~,Q)1)IIflib(Q), withCdependingonlyonn.

    Proof.Thefirstinequalityisfollowedbythedefinitionofh(Q).Forthesecondinequal

    ity,chooseanonnegativetestfunction0?()suchthat001,0=1inQand

    0=0outsideQwithlDOI4(dist(Q,Q))?.DirectcalculationshowsthatlIelICl(Rn) 360Z.Sunetal:SolvabilityinHardySpaceforSecondOrderEllipticEqua~ons

    C(1+Iln(dist(f~',Q))I).Suppose?h1(gn)isanyextensionoffintORn.Bythecondition suppfccQccQ.wehave

    of=finR.

    ByProposition2.1onehas

    ,-()=I10fllh()cIIOllc,(R"/llfllh-()c(1+Iln(dist(~,0a))1)llY[I^-()

    Thesecondinequalityisthenfollowedbydefinition. Lemma2.2.(l4,Lemma5.2)LetQbeaboundeddomaininR.Iff?(Q)withq>1,

    thenf?h(Q)and

    IIflib(Q)C(n,q,Q)Il'll(Q).

Thenextlemmaiscrucialintheproofofboundaryestimate.Denotebytheupperhalf

    space.Forafunctionfdefinedin,bydenotingX=(,),letx)=,),?R~;fo(X)=

    

    f(xt,),?R.foisanoddextensioninthelastvariableXnoff. Lemma2.3.1ff?h.(R),thenfo?h(R)with

    whereCisindependentoff

    fllh-(R)lIfollh()Cllfllh(),

    ThislemmaCanbeprovedbyatomicdecomposition.Notethatforatype(a)atom(re

    spectivelytype(b)atomwithr1)in,thereflectionpartinisalsoatype(a)atom

    (respectivelytype(b)atom).Foraboundaryatomb(x)withsupportingballBr(x0)c,r<1,

    notethatbo(x)isatype(a)atomwithsupportingball,forexample,B10r(jc1),whereX1=(,0

    ).

    FormoredetailsonecanrefertOSection1in[5.

    3EstimateforLwithConstantCoefficients

    InthissectionwegivetheestimateinHardyspaceforLaplacianoperatorAandoperator

    Lwithconstantcoefficients.AlthoughtheresultforLaplacianisknown,theprocedurehereis

    differentfromthosein[4,5].Hencewegiveadetailedanalysis. Proposition3.1.LetQbeboundeddomaininR,?'(Q),andAu=finQ.If

    f?h1(R)withsuppfcQthenD?h1(Q)andthereis?constantC=C(n)suchthat

    Dullh-(Q)CIIfllh(R)(3.1)

    Ana1.TheoryApp1.,Vo1.25,No.4(2009)361

    Proo~Forf,J=1,2,,n,sincehascompactsupportinQ,wehave

    Di"(),():JRnFij(Xy)f(y)dy,a?e?xEQ?(3?2)

    Herer(x)isthefundamentalsolutionofLaplacianoperatorandFij(x)=DijF(x)isakernel

    ofstandardCalder6n

    Zygmundsingularintegraloperator.NotethatDiju(x1isdefinedfora.e. ?R.Sincef?h(R),byL()andH(R)boundednessofCZsingularintegral,one

    immediatelyhas

    IIT0alIH(n)C,

    ifaisanatomoftype(a),and IfzC,

    ifbisanatomoftype(b)withtheradiusofsupportingballnotlessthanone.Hencebythe

    atomicdecomposition,

    IIDijull,(R)+L2(Rn)CI]fllh(n). ByLemma2.2andarestrictionargumentoneconcludestheproof.

    TodealwiththeoperatorL=r.in=laijDijwithconstantscoefficient,wefirstgivethefol lowing

    Lemma3.1.For>1andf?L(R)define

    ,,z?ot,()supl,()1.

    0<f<

    Then

    IIm~fl1.(n)II/llh-(

    Pro0fAdirectCalculationshows Thereforeweonlyneedtoprove ,,z)=(,(?))().

    f(a')tlh(R)IIfllh-(Rn).

    aILh()1,lla(a(?X0))llh().

    Similarargumentappliedtoeitheratype(a)atomwithr1/aoratype(b)atomb(x)shows

    thatbisanatomoftype(b)andhence b,llh-()1,lIb(a.)lib-()a

    -!

    'g

    w?

    ?

    

    .?m{l

    

    "

.

    li_

    m

    ?

    ?

    

    .uIl

    ?a

    bn

    b<

    {

    .

    T8

    362YZsunels.l.abilityinHafd

    yspfdcef0fSecondomerEllipticElti1

    )nI;

    Theorem3.

    1

    1.jDjconstantco

    g

    (?2)?1fuE2(Q),fEh(R)wsuppfc12 :,Q,

    ItD2/~II^t(Q)<_ Cllf1]n),(3.

    3)

    wtheconstantCdepends.

    ,zty册力,,.

    f0nnD

    Theproof

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