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SOME ESTIMATES OF MAXIMAL COMMUTATORS FOR BOCHNER-RIESZ OPERATOR ON MORREY SPACES

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SOME ESTIMATES OF MAXIMAL COMMUTATORS FOR BOCHNER-RIESZ OPERATOR ON MORREY SPACES

    SOME ESTIMATES OF MAXIMAL

    COMMUTATORS FOR

    BOCHNER-RIESZ OPERATOR ON

    MORREY SPACES

    Ana1.TheoryApp1.

    Vo1.26,No.1(2010),7683

    DoI10.t007/s1049601000761

    SoMEESTIMATESoFMAXIMALCOMMUTATORS

    FoRBoCHNER.RIESZoPERAToRoNMoRREYSPACES

    LingliHuandDongxiangChen

    (JiangxiNormalUniversity,China)

    ReceivedSep.14,2009

    ?EditorialBoardofAnalysisinTheory&ApplicationsandSpringer

    VerlagBerlinHeidelberg2010

    Abstract.Inthisnote,theauthorsshowtheboundednessofthemaximalcommutators

    ofBocherRieszoperatorBandthatofmaximalcommutatorB3,generatedbyBanda Lipschitzfunctionbmappingfrom()intoBMOspaceandalsomapsfrom(Rn) intoL(卢一)?

    Keywords:Bocher-Rieszoperator,commutator,Lipschitzfunction AMS(2010)subjectclassification:42B20,42B25,42B35 1IntrOductiOnandMainResults

    Itiswel1.knownthatacommutatorisanimportantintegraloperatorandplaysakeyrolein

    harmonicanalysis.In1965.Calder6n[1J12Jstudiedakindofcommutators,

    whichhappenedin

    CauchyintegralproblemsofLipline.Subsequently,Coifman,RochbergandWeissjJobtained

    theboundednessofsingularintegralcommutatorsonLP(R)(1<P<0o).In1996,Luand Hu4]f5

    establishedtheboundednessofBochne~RieszcommutatorsonL(R)(1<P<o.). In1997.YangandLu[6JstudiedthecontinuityofBochner-RieszcommutatorsonHerz

    type

    spaces.In2003,LuandLiu[7JestablishedtheL(LogL)typeestimateandweightedweaktype estimateofBochner

    Rieszmaximalcommutators.In2004.Liu[establishedthecontinuityof Bochner-RieszmaximalcommutatorsonTriebelLizorkinspaces.In2007.Lint?

    Jstudiedthe

    boundednessofstronglysingularCalder6n

    ZygmundoperatorsandcommutatorsonMorrey

    typespaces.Themainpurposeofthispaperistogivesomeendpointestimatesofmaximal commutatorsofBocher-RieszoperatoronsomeMorreyspaces.

    SupposedbyNNSF(10961015,10871173),JXNSF(2008Gzs0051),GJJ08169) Ana1.TheoryApp1,Vol26,No,l(2O1o)77

    Defniti.n1.1.Let()=(1-t21~])6()andBf(z)~_t-n().F.rf>0,wede

    n0te

    83.(,))=faH()一易(y)]B)'),(),)d

    ThemaximalBochner-Rieszoperatorandmaximalcommutatoraredefinedrespectivelyby and

    Defnition1.2.Thefunctionf?Leoc(R)issaidtobelongtotheclassical(R)space,

    1Pq<oo,if

    ]lfll~(Rn)=l(Llf()p1)1<c..

    Defnition1.3.If0<<1,theLipschitzspaceconsistsoffunctionssatisfying f//Lip/3=

    ,

    s

    .

    u

..

    p

    ,?.

    :..

    The.rem1?1.Lbefore,ELip/3(R),0<<1?n<p<oo

    卢一<1,卢舭倒istapositiveconsf

    (f)(x)]IBMOCllbllciIlfli~(R), g.

    The.rem1?2?LetB~,,zPbefore,n?Lip/~(R)?n<p?口<?,n

    thereexistsapositiveconstantCsuchthat

    3,()llLip(pg)Cllb[1Lipt~Ilfll~(Rn) 2ProofofTheorems

    BI+lVB()lC

    (1+)6+

    Lemma2.2.LetQ=rJdenotethecubewithcenterXandlength,:Thenforally>

    0,P>1,wehave

    az?c

    ,,,

    /,l

    \/

    

    ,,

    r

    p0U>

    Sf

    lI

    ,?/

    ,,

    ,?,

    f,,,l ,

    IJ

    /

    ,,, ,,

    /,

    pOU>

    Sf

    =

    ,l, /,

    ,,, f,/, w

    sm

    n

    L

    m

    .J

    ef

    w=

    一耵

    .

    m

    

    

    78L.L.HuetalSomeEstimatesofMaximalCommutatorsforBochner-RieszOperator

    ForanyballB=B(xo,r),lethQ=B(((易一bB)fzR\2B)(x0)?InordertoproveTheorem

    1.1.itsufficestoshowthat

13,

    ()()-hoIclJLII(R

    NOWweconsider

    1(,)()-hQI

    =

    ((((bB)fz()(I

    )J(_)()+f((b-bB)fz(z))() +lts(((b-bB)fz()c))()ts(((b-bB)fz(zB)c))()l Wenowestimate,1.TheH61derinequalityandtheLn

    boundnessofB,8

    ,

    imply

    ,clIL,-)(

    cLiprfl/

    ~?l5)

    cllbllLip.l(p)

    /,,1

    f/lI(pdxl\J

    B/

    CllbllLipp()

    ByusingthesameargumentasI1,wehave

    /2C[[b[1Lipp()

    For/3,weconsiderthefollowingtwocases:

    Case1.0<t<Inthiscase,forany?B,Y?(2B)c,wehave 2.1.themeanvalvetheoremandH61derinequalitytellUSthat

    

    Y『?IXOy1.Lemma

    ,3.1

    .fs

    f>0

upf

    B

    [.),)_Bt5(加一y)llIrf(y)ldydx cLi]~]fBsupfB

    Cllbll(n11,,t,6

    up

    0r

    ()'(y)

    f

    J22B

    y

    Ana1.TheoryApp1.,Vo1.26,No. J(2010)

    2k(<Tx-a)}2k+lBI

    cfffILi:i(+f(y)I)

    c()

    llbllLioIlfll~(g,,). C

    79

    case2?>nthiscasewech..sesuchthat<<min(, ).Lemma2.1'themeanvalvetheoremandH61derinequalityimplVz

    ,3cfLB,t-(n+1)l(-+)6n+l' cIJ6JJLjp()一面,.面一』,+.\:If(Y_=)lly-xo18dy

    cffL'12k+lBlg

    CllbllLip(高咖)

    ()

    <CfL1(p)

    C'llbllLipIlfli~(R).

    Therefore,weobtain

    f2,(-))--hofcIILIlfll(R

    ThiscompletestheuroofnfThPnTpm1

3ProofofTheorem1.2

    InordertoproofTheorem1.2,weconside

    3,(,)()?3.

    ())I

    IB3,((2))()J+J3,((2))) 8OL.L.HuetalSomeEstimatesofMaximalCommutatorsforBochner-RieszOperator

    +

    

    (?z)])dz

    IB6

    JR())6(z))dzI\2Bt\/L\,JJ\,.?l

    :I+J+K.

    ,s

    >

    up

    .

    f

    B

    J(L~-Y-)llb(x)——(z)llf(z)jaz

    II厶孚)zIf(z)laz.

    Weconsiderthefollowingtwocases: Case1.0<t<r.Iniscase,foranyX?B,Y?(2B)c,wehavelXYl?IX0y1.Lemma

    2.1,

    .

    Lemma2.2,H61derinequalityand(+字一)("-v-+dIi--ud1)----,,tell

    ,c(~n-

    I厶出

    cIIblJup~r#-nIJ

    =CIIbllLipar~-nlgl~?1I

    =

CIIbllLipar~.

    Case2.f>r.Inthiscasewech..se50suchthatn-1 <<min(,),thenby

    Lemm

    

    a2.1,Lemma2.2theH61derinequalityand(+一卢)pt=n-(卢一n+n-1一面)p

    ,ffffLi,.(南一'()面一.

    Lf(z

    

    )]dz

    CllbllLip~F[3-nff =Cl[bllLip~rtJ-ni11 =

    cLip,.卢一厂ll.

    Bythesameway'wehavecLipp,_卢一厂ll. WenowestimateK.Wte

    \?)dzJ

    +6(X--Z)()]()dzI

    Ana1.TheoryApp1.,Vo1.26,No.jr2fJJ

    FirstweturntodealwithK1 -cIIbllLipsup

    .R,\2BB6()dZI

    clps

    >

    up

    n

    f?

    fRHtJRn\2B1)))Idz>O lLi

    t>0

    2B)

    

    (/d?(L\

    SimilartoI,wedealwithK1intwocases.

    Case1.0<t<Inthiscase,foranyX?B,Y?(2B)c,wehave XYI?lX0Y

    Lemma2.1,themeanvalvetheoremandH61derinequality,tellUS

    

    yIr6(;)-~-oo厶曰

    clIL

    cllIILiIll(+.I,(z)Ipdz) clllILipl),I

    2

    

    k:B二虿(+I(y)Ip)l+ll亩一嚣\2+/

    Il/ll~

    n1

    1

    l

    n

    II/ll~.

    If(z)

    

    l

    8l

    case2.f>Inthiscasewech..sesuchthat<<min(,),thenby

    Lemma2.1,themeanvalveTheoremandH61derinequality,weobtain

    ?cllLi即一),};()n-I-~04-00+B

    If_=zI(

    z)

    +

    l.

    dZ

    cIlbllLipI卢南If(z)laz

    ?c1I6IILipIyll芋量(+I_(z)Ipdz)

    c?--Lip?—一卢赤(+tcyp)

    ...

    

    CCC

    <==

    82L.L.Huetal:SomeEstimatesofMaximalCommutatorsforBochner-RieszOperator

    Asfor,wealsohave

    nIIfll

    in11J

    n

    Ilfll~.

    lfLn()(z)

    Weconsiderthefollowingtwocases: When0<t<forany?B,Y?(2B)c,wehaveYl?IX0y1.Lemma2.1,themean

    valvetheoremandH61derinequalitytellUS ^,2cIllILi,.;8()+,:az

    CllbllLipIf(z)ldzp?2.

    (f22~+1B]f(y)Ipdy)n

    Cllbl[LipalIfll~

    =cLippnl一厂fI

    :CllbllLipanIIfll~.,,

    Whenf>,.,wechoose6osuchthat<6o<min(,),thenLemma2.1,themean

    valvetheoremandH61derinequalitytellUS

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