Transference of bilinear multiplier operators on L

arXiv:0709.4339v2[math.CA]30Sep2007AMSClass(2000)42A45TRANSFERENCEOFBILINEARMULTIPLIEROPERATORSONLORENTZSPACES.OSCARBLASCO,FRANCISCOVILLARROYAAbstract.Letm(ξ,η)beaboundedcontinuousfunctioninIR×IR,0pi,qi∞fori=1,2and0p3,q3≤∞where1/p1+1/p2=1/p3.ItisshownthatCm(f,g)(x)=ZIRZIRˆf(ξ)ˆg(η)m(ξ,η)e2πix(ξ+η)dξdηisaboundedbilinearoperatorfromLp1,q1(IR)×Lp2,q2(IR)intoLp3,q3(IR)ifandonlyifPDε−1m(f,g)(θ)=Xk∈ZXk′∈Zˆf(k)ˆg(k′)m(εk,εk′)e2πiθ(k+k′)areboundedbilinearoperatorsfromLp1,q1(T)×Lp2,q2(T)intoLp3,q3(T)withnormboundedbyuniformconstantforallǫ0.1.Introduction.Letm(ξ1,ξ2,...,ξn)beaboundedmeasurablefunctioninIRnanddefineCm(f1,f2,...,fn)(x)=ZIRnˆf1(ξ1)...ˆfn(ξn)m(ξ1,ξ2,...,ξn)e2πix(ξ1+ξ2+...+ξn)dξforSchwartztestfunctionsfiinSfori=1,...,n.Givennow0pi≤∞fori=1,...,nand1/q=1/p1+1/p2+...1/pn.Thefunctionmissaidtobeamultilinearmultiplierofstrongtype(p1,p2,...,pn)(respect.weaktype(p1,p2,...,pn))ifCmextendstoaboundedbilinearop-eratorfromLp1(IR)×...×Lpn(IR)intoLq(IR)(respect.toLq,∞(IR)).ThestudyofsuchmultilinearmultiplierswasstartedbyR.CoifmanandY.Meyer(see[4,5,6])forsmoothsymbols.However,inthelastyearspeoplegotinterestedinthemaftertheresultsprovedbyM.LaceyandC.Thiele([21,22,23])whichestablishthatm(ξ,ν)=sign(ξ+αν)aremultipliersofstrongtype(p1,p2)for1p1,p2≤∞,p32/3andeachα∈IR\{0,1}.Newresultsfornon-smoothsymbols,extendingtheonesgivenbythebilin-earHilberttransform,havebeenachievedbyJ.E.GilbertandA.R.Nahmod(see[10,11,12])andbyC.Muscalu,T.TaoandC.Thiele(see[20]).Wereferthereaderto[18,17,9,13]forseveralresultsonbilinearmultipliersandrelatedtopics.BothauthorshavebeenpartiallysupportedbygrantsDGESICPB98-0146andBMF2002-04013.12OSCARBLASCO,FRANCISCOVILLARROYAThefirsttransferencemethodsforlinearmultipliersweregivenbyK.Deleeuw.ItisknownthatifmiscontinuousthenTm(f)(x)=ZIRˆf(ξ)m(ξ)e2πixξdξ(definedforf∈S(IR))isboundedonLp(IR)ifandonlyif˜Tmε(f)(θ)=Xk∈Zˆf(k)m(εk)e2πiθk(definedfortrigonometricpolynomialsf)areuniformlyboundedonLp(T)forallε0(see[8],[29]page264).Althoughtheresultsinthepaperholdtrueformultilinearmultipliers,forsimplicityofthenotationwerestrictourselvestobilinearmultipliersandonlystateandprovethetheoremsinsuchasituation.Let(mk,k′)beaboundedsequenceweusethenotationPm(f,g)(θ)=Xk∈ZZXk′∈ZZa(k)b(k′)mk,k′e2πiθ(k+k′)forf(t)=Pn∈ZZa(n)e2πintandg(t)=Pn∈ZZb(n)e2πint.Let0p1,p2≤∞andp3suchthat1/p1+1/p2=1/p3.WewritePDt−1mwhenthesymbolism(tk,tk′)andsaythatm(tk,tk′)isaboundedmultiplierofstrong(respect.weak)type(p1,p2)onZZ×ZZifthecorrespondingPDt−1misboundedfromLp1(T)×Lp1(T)intoLp3(T)(respect.Lp3,∞(T)).Inarecentpaper(see[9])D.FanandS.SatohaveshowncertainDeLeeuwtypetheoremsfortransferringmultilinearoperatorsonLebesgueandHardyspacesfromIRntoTn.Theyshowthatthemultilinearversionofthetrans-ferencebetweenIRandZZholdstrue,namelythatforcontinuousfunctionsm(ξ,η)onehasthatmisamultiplierofstrong(respect.weak)type(p1,p2)onIR×IRifandonlyif(Dε−1m)k,k′=(m(εk,εk′))k,k′areuniformlyboundedmultipliersofstrong(respect.weak)type(p1,p2)onZZ×ZZ.Thefirstauthor(see[1])hasshownaDeleeuwtypetheoremtotransferbilinearmultipliersfromLp(IR)tobilinearmultipliersactingonℓp(ZZ).Theaimofthispaperistogetanextensionofthoseresultsin[9]forbilinearmultipliersactingonLorentzspaces(see[9],Remark3).WeshallshowthatifmisaboundedcontinuousfunctiononIR2thenCmdefinesaboundedbilinearmapfromLp1,q1(IR)×Lp2,q2(IR)intoLp3,q3(IR)ifandonlyifthePDt−1m,therestrictiontom(tk,tk′)fork,k′∈ZZ,definebilinearmapsfromLp1,q1(T)×Lp2,q2(T)intoLp3,q3(T)uniformlyboundedfort0.Throughoutthepaper|A|denotestheLebesguemeasureofAandweidentifyfunctionsfonTandperiodicfunctionsonIRwithperiod1definedon[−12,12),thatisf(x)=f(e2πix)andRTTf(z)dm(z)=R12−12f(t)dt.For0p≤∞,wewriteDptf(x)=t−1pf(t−1x)(withthenotationDt=D∞t),TRANSFERENCEOFBILINEARMULTIPLIEROPERATORSONLORENTZSPACES.3Myf(x)=f(x)e2πiyxandTyf(x)=f(x−y)forthedilation,modulationandtranslationoperators.Inthisway(Dqtf)ˆ=Dq′t−1ˆfwhere,asusual,q′standsfortheconjugateexponentofq.Adknowledgement:Wewanttothanktherefereeforhisorhercarefullreading.2.PreliminariesLet(Ω,Σ,μ)beaσ-finiteandcompletemeasurespace.Givenacomplex-valuedmeasurablefunctionfweshalldenotethedistributionfunctionoffbyμf(λ)=μ(Eλ)forλ0whereEλ={w∈Ω:|f(w)|λ},thenonincreasingrearrangementoffbyf∗(t)=inf{λ0:μf(λ)≤t}andf∗∗(t)=1tRt0f∗(s)ds.NowtheLorentzspaceLp,qconsistsofthosemeasurablefunctionsfsuchthatkfk∗pq∞,wherekfk∗pq=qpZ∞0tqpf∗(t)qdtt1q,0p∞,0q∞,supt0t1pf∗(t)0p≤∞,q=∞.Itiswellknownthat||f||p∞=supλ0λμf(λ)1/p.Hereweshallusethefollowingfact:If0p,q∞andfisameasurablefunctionthen(1)kfk∗pq=qZ∞0λq−1μf(λ)qpdλ1/q.(Thiscanbeeasilycheckedforsimplefunctions).Letusrecallsomefactsaboutthesespaces.SimplefunctionsaredenseinLp,qforq6=∞,(Lp,1)∗=Lp′,∞for1≤p∞,and(Lp,q)∗=Lp′,q′for1p,q∞aswell.Replacingf∗byf∗∗andputtingkfkpq=kf∗∗k∗pqthenwegetafunctionalequivalenttok·k∗pq(for1p∞)forwhichL1,1andLp,qfor1p≤∞,1≤q≤∞areBanachspaces.Thereaderisreferredto[19],[2],[29]or[25]forthebasicinformationonLorentzspaces.WeonlycondiderμtobeeithertheLebesguemeasureonIRorthenormalizedLebesgemeasureonTandthedistributionfunctionwillbedenotedmfinbot