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DOUBLE

    DOUBLE

    Chin.Ann.ofMath.

    21B:2(2oo0),211-216

    DoUBLE圣一FUNCTIoNINEQUALITYFoR

    NoNNEGIVESUBMARI'INGALES,lc,lc

    MEITAoLIUPEIDE*

    Abstract

    Theauthorsestablishakindofinequalitiesfornonnegativesubmartingaleswhichdepend

    ontwofunctionsand.andobtaintheequivalentconditionsforandsuchthatthiskind

    ofinequalitiesholds.Intheca.Be=??2,itisprovedthatthisnecessaryandsufficient conditionisequivalenttoq>1.

    KeywordsMartingale,Nonnegativesubmartingale,Maximalfunction,function

    inequality

    1991MRSubjectClassification60G42.46E30

    ChineseLibraryClassificationO211.6,O174.13DocumentCodeA ArticleID0252?-9599(2000)02?-0211?-06

    ?1.Introduction

    Letbeanonnegativenondecreasingcontinuousfunctionon[0,oo)with(0)=0and .1im(t)=oo,and(Q,?,)beacompleteprobabilityspace.Wedenotebytheset ofall?一measurablefunctionsandL(Q)={,?,e>0,EO(clf1)<o.),whereE standsfortheexpectationwithrespectto,L(Q)iscalledanOrliczspace.Infact,it isakindofspacemoreextensivethanclassicalOrliczspace.Whenisconvex.wedefine

    thenormonitbyllfII=inf{k>0,E()1).Let?n?1)beanondecreasing

    sequenceofcompletesub-afieldswith?=V?nanddefinemartingaleorsubmartingale f=(,n)>0asusua1.Denotethemaximalfunctionoffbyf()=supI^()I.As wellknowninmartingaletheory,whenisastrictlyconvexfunctionon[0,oo),i.e.=

    inf

t>0

    >1(whereistherightcontinuousderivativeof),thefollowinginequalities hold:(,)?supEO(cf,,),foreverynonnegativesubmartingalef=(^)n>0,wherec n>0——

    isaconstantonlydependingon.Whenisnotstrictlyconvex.thesituationisvery different.Toseethis,weonlyneedtorecallDoobSinequalityinthecaseP=1, Ef+<

    e1(1+supEn>0l^llog+l^l

    Thatistosay.fisinLwhenf?Llog+L.ThisinspectinspiresUStoconsidermaximal functioninequalitiesrelatedtotwofunctionsand.

    ManuscriptreceivedMarch8,1999.RevisedDecember8,1999.

    *CollegeofMathematicsandComputerSciences,WuhanUniversity,Wuhan430072,China E-mail:meitaosuizhou~263.net;pdliu~whu.edu.ca

    **ProjectsupportedbytheNationalNaturalScienceFoundationofChina.

2l2CHIN.ANN.OFMATHVo1.21Ser.B

    Suppose,aretw.n.nnegativenondecreasingc.ntinuousfuncti.nsdeed0n..

    withf0):(0):0,andisconvex,i(?)=??Through.upp,,

    alwavsdf0rerightcontinu.usderivatiand,respectiVely.Weshallc0ndsider thefollowingconditionaboutand,

    ds(ct),Vt>0J0S

    Wewillprovethat(1.1)isequivalenttoanyoneofthe--

    f0ll0Wingcondo

    .,r-

    fi1Thereexistsc2>0suchthatE~2(f)su,

    pEke(c2,n)foreVeynonnegatVeub

    ,,U

    mn>O-

    exists0suchthatE()clEg~()ro……Y……give.n(ii)Therec1>E(JIJV.nlIlenu

    pair{gwhichsatisfies

I{,>)I

    ,>^)

    9d,V>.,(1.2)

    where0<,<?.Moreover,inthecase=?A2(i.e.theresap0sltlV: constaIItcsucthat(2)c()forallt>0),weprovedthatCondition(1?1)isequVent

    to>1.

    ?2.TheMaximalInequalities ?L(f1)uL().Then

    =

    >Vt>0

    Lemma2.2.Let,:(,n)0be?nonneg?tiuesubmarting?le?Thn Vt>0,n??

    Kolmogoroflsinequality,wehave.. >l{>)l+i{^>t)lh,~dtz=2>(,n一三)d (2.1)

    (2.2)

    n

    

    c

    l1

    eu0ws2J

    皿几s删咖Th21Su1.1

    pposeandaret1uCI,UUThe0rem..enc0.5u……'

    equivalenttoanyoneofefollowingstatem.t:

    fi1The'reexistsC2>0suchthat

    E(,)supEk0(c2,n)(2?4)

    reuenonneg?ti"esubm.rtin9ale,=(,n)no' d?帆

    Am

    

    

    m

    m

    mm

    hk

    

    

    岫嘲

    A

    2tm

    

    ?n??唧

    N0.2MEI,T.&LIU.P.D.DOUBLEFUNCTIONINEQUALITY213 (ii)Thereexistsel>0suchthat E()c()(2.5)

    ,0reverynonnegativenctionpairf,gsatisfying(1.2).

    Proof.(i)Toprove(1.1)(2.4),noticethat(2^)>0isanonnegativesubmartingale

    andreplacet,fin(2.2)by2t,2f,respectively.Thenintegrateitsbothsideswithrespect

    tod(),andweget

    rooroo1roo

    EO():/l{2>2t}[dO(t)/?/l{2^>A}[dAdO(t) :

    1{2^>)Jd(?)d/ol{2^>)Ic(c)dA=EO(2c^) Lettingn_?oo,weget(2.4).

    Nextweprove(2.4)(1.1).Considerthefollowingdyadicmartingaleon(0,1:Let

Ak=(1一去,1,=a{A1,A2,?,Ak},f=tXA,wheret>0,k,n?N,anddenote

    ,:(E(II))k0.Then,isafinitemartingalewithfk:f(kn). Itisclearthatlfklt(VkO),thusft.Bytheconvexityof,weget EO(e2f):EE(q2(c2f)[.Tk)EO(E(c2f[.Tk))=E(c2fk),k0 andthensupEO(e2)=EO(e2f).From(2.4)wehaveEO(f)EO(e2f),i.e ,?,?

    /l{,>s}l~(s)ds/l{a2f>s}l~(s)ds.',0J0

    Noticethatwhens?(,)(okn1),wehaveI{f>s)

    <T2n--k

    ,SO<竿.=竿?l{广>ss?(.Therefore

    and

    

    2n+l

    l{f)ds竿..l{cz,>s)ds?

    Nowfromftweget

    2n+1~0l{c2f>s?莩)ds<2c2~(c2

    Lettingn_?OO,we.btainds2c2(c2t),V?>0.Thispr.ves(1.1).

    (ii)Toprove(1.1)(2.5),weintegratebothsidesof(1.2)withrespecttodO(A),and get

    E()..I{f>)ld()..,>

    gd#dO()

    =9

    

    9

    (c)

    Thisis(2.5).

    Toprove(2.5)(1.1),let,:(^)0bethedyadicfinitemartingaleaLsintheproof of(i).Thenthenonnegativefunctionpairf,^satisfies(1.2)with=1,=1?Hence

    (2.5)holds,thatistosayEO()c1E^(c1).Fromthisandtheproofof(i)weget

ds<)竿cEc)c

    竿c)T.1.J2c1~(clt)e~(et)

214CHIN.ANN.0FMATHV01.2lSer.B

    Letting几?,wehavedsc(ct).ThenTheorem2.1follows. Theorem2.2.Supposeandnthefunctionsmentionedabove.Thenthecondition that,satisfy

    ,0rsomeSO,Cl>0,isequivalenttoanyoneofthe,0llowingstatements:

    (i)Thereexistc,c2>0suchthat

    EO(f)sup[cE,n+E(c2,n)]

    n

    ,0reverynonnegativesubmartingale,=(,n)n0. (ii)Thereexistconstantsc,Cl>0suchthat

    E()cEg+cl()

    ,0reverypairofnonnegativeyunction,,gsatisfying(1.2). Proof.(i)Toprove(2.6)(2.7),fromFubinitheorem,weget E()=/l{2>2t}ldO(t)+/l{2>2t}ldO(t).,80J0 ..

    f{2,n>}ldAdO(t)+..f{2,n>)ldd(t)

    _I{2,n>)I{d(t)d+..I{2,n>)I{d(t)d

    /l{2,n>A}ICl~(C,A)dA+L/l{2,n>A}IdA

    E(2cl,)+2LE,n.

    (2.6)

    (2.7)

    (2.8)

    Let-??,then(2.7)follows.

    Toprove(2.7)(2.6),bythediscussionsimilartotheproofofTheorem2.1,wecanget

    l

    _+imdsE()2c+2c2~(2c2t).Denotes.=1+inf{t>0,(2c2t)>0). ThenVt>0,

ds2c+2c2(2c2t)Js0s

    .

    fds

    2c

    ~(2c2s0)

    2c

    ~(2c2s0)

    +2c21(2c2t)

    d

    S

    ~(2c2s0)+2c2~(2c2t)

    cl(clt),

    2c+2c2~(2C2So),

    whichisdesired.

    (ii)Byanargumentsimilarto(ii)ofTheorem2.1,wegetthat(2.6)isequivalentto(2.8)

    Eventuallywegetthefollowingresult.

    Corollary2.1.Ifthefunctionsandareasabove,thenthecondition(2.6)

    equivalenttothestatementthatthereexistsc5>0,0rany>0,suchthatEO(f)

    sup[SEfn+E(cdn)],0reverynonnegativesubmartingale,=(,n)0. n

    ?3.TheDiscussionAboutCondition(1.1)andSomeExamples Inthecase

    7,S

    <

    .

    Theorems2.1and2.2becometheclassicalmaximalfunctionin

    

    h

    No.2MEI,T.&LIU,P.D.DOUBLE4-!!!!!215 equMitiesforn.nnegativesubmartinga1e.Thefoil.wingtheoremgiVesanexactresultund

    '

    ..

    圣?funct0n『一,,.ThThorem31LetA2beanondecreasingconuextUoo)e>1e..

    圣?00n0n,''q,

    一如c1>O,sd'Clt)

    

    ,

    Vt

    n

    >0

    ProLiWefirstprovethenecessityNoticethatthcondxt10J0tds<(c1tt)

    o().e.5c1,

    phes(sJ,0(aSs0)andds<..,andfrom西(?)(s)ds()wg inf

    t>O

    Then

    >1.Denote

    Therefore

    .=2k(2)(2){2k-I(2)一西(2七一)(.o<k<+..)

    2一【(2)(2)】?k2k(2)(2),

    (2)

    ?2k.(2)(o)<.?2+.k

    =oon

    2m1f2

    

    /2

    Ontheotherhand

    Noticethatinf

?>0

    3tj?

    +1

    22一件0i

    m1

    (s)ds?2(2?)=

    i=oot=oo

    1

    >

    2

    厂?ds>2S.,2k='

    :ooi=oo

    (2,2j+suchthat 2n~-k+lak)

    Thenforko=njJ+1, 0

    1

    

    2

    m1

    ?2(仇一i).i (3.1)

    (3.2)

    (3.3)

    (3.4)

    (3.5)

    :1,and(3.1),(3.4)implythefact:>1,

    2",(2j)一西(2"j) 2n广川.).1,= b+1

    Thus22 =..

    Thenecessityfollows.

    =o0

    2-ka, and

    (2") ko

    (?:

    (tj)() ()

    2~J--k+lak)nj?

    =oo

    1

    <?

    2,

    1

    0k<-

    ~2.

    i-1.?

    

    0

    

    2?

    一?

    妻一

    妻一知,,.,,

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