广义Littlewood—Paley函数的BMO有界性

广义Littlewood—Paley函数的BMO有界性 广义Littlewood—Paley函数的BMO有界性 JournalofMathematicalResearch&Exposition Vo1.22,No.4,515—518,November,2002 BMOBoundednessofGeneralizedLittlewood—Paley Functions LIWen—ruing (Dept.ofMath.,HebeiNormalUniversity,Shijiazhuang050018,China) Abstract:ItisprovedthattheimageofaBMOfunctionunderthegeneralized Littlewood—PaleyflmctionsiseitherequaltoinfinityallnosteverywhereorinBMO. Keywords:Littlewood-Paleyfqnction;BMO. Classification:AMS(2000)42B25/CLCO174.3 Documentcode:AArticleID:1000-?341X(2002)04-0515??04 For?R"andt>0,thePoissionkernelfortheupperhalfphane,R,is P(,t):c thePoissionintegralof,()?}0(R")is (t+ll)' ,(,t)=[,P(?,t)]()=/P(—Y,t)f(y)dy.,Rn andthegradientof,(,Y)is V'z)=(…,芒of. TheLittlewood-Paleyfunctiong(f)isdefinedby g(,):(厂tlVf(tlVf(w,t)ldt)}.g(,)=(/,dt);.J0 Wang[1provethatforf?BMO(R"),eitherg(,)()=?almosteverywhereorg(,)()< ?almosteverywhereandthereisaconstantCdependingonlyonnsuchthat g(f)llCllfll, 'Receiveddate:2000-01.17 Biography:LIWeu—ming(1963一),llla]e,Ph.D.,AssociateProfessor —--—— 515---—— whereIl?isthenorminBMO.Inthisnote,wewillgeneralizethisresulttoamore generalcase. Let (1) (z)isfunctiondefinedonR",for>0and7>0,satisfying l(z)l矸,z?R" (2)foranyz,Y?R",lz—l(1+lz1), ))p(1 +l1)"+' (3)()dz=0. WedefinethegeneralizedLittlewood—Paleyfunctionby G(,)():(厂If木()T)-l,G(,)()=(/木t()…,J0 where,(z)?}0("),t()=专() Lernma1Letf?BMO(R"),7>0andP1,letQbeacubecenteredatzandhave edgelengthr.ThereisaconstantCdependingonn,7andPSOthatfort>0 (f(Y)一fQlp (1Y—l+,)"+ 1 Ct一3(1dy)FCt+l1Il[lIfll一+l1Il[二]1)l TheproofofthisLemmaissimilartothe result. Lemma1.1in[2].ThefollowingisOUrmain Theorem2Letf?BMO(R").EitherG(,)()=O0almosteverywhereorG(,)()<O0 aln]osteverywhereandthereisaconstantCdependingonlyonnsuchthat ProofSupposeG(,)()? positivemeasure.Let孟bea set=南f(t)dt.Write C(f)llcllfll. O0almosteverywhere.TheE pointofdensityofE,andQbe fas = {:G(,)(z)<?].has anycubecenteredatand ,()=.+[,()一.Ix0(.)+[,()一.]Qr()=.+gQ(x)+^Q(z), whereQdenotethecomplementofQ.Since ThusG(,Q)isinBMOwithBMOnormequal and isaconstant,G()isidentically0. to0.Therefore, G(f)G(gQ)+G(hQ) G(hQ)C(f)+c(go). Sincef?BMO(R"),wehave gQll2=(f(t),Qld,)ClQl~llfll 一 516— andg.?L.Thus,G(gQ)isfinitealmosteverywhere.Therefore,G(厂)(z)<ooataltnost everypointsuchthatG(hQ)()<?. Letd<1.SinceisapointofdensityofEandG(gQ)isfinitealmosteverywhere, thereisapointzindQsuchthatG(厂)(,)1G(gQ)()anda(hQ)(x)arefinite. InthefollowingLemma3wewillprovethatforasufficientlysmalld,thereisaconstant CSOthatforallz?dQ, (i)G(hQ)()<?G(hQ)()<?, (ii)JG(hQ)(z)一G(hQ(,)lllfll. NOWweassumethat(i1and(ii1aretrue.FixacubeQcenteredat奎.Asabove, thereisan?dQSOthatG(hQ)()<...By(i),G(hQ)()andG(厂)()isfinite almosteverywhereindQ.Consideringonlycubescenteredat奎withedgelengthequalto apositiveintegershowsa(f)isfinitealmosteverywhere. NowweprovethatJIG(/)IIJJ,.LetQbeanycubeandsetQ=Q.Choose apoint?dQSOthatG(hQ)()isfinite.Thenby(i)and(ii), G(,)()一G(hQ)(x)lax G(gQ+hQ)(x)一G(hQ)(x)+a(hQ)(x)一a(hQ)(x)lax Q)(d+1 , IG(()一G((Ilfll? SoJIC(/)llCllfll,theproofiscomplete Lennna3Supposef?BMO(R"1.LetQ r.Setd=1. Supp.secJ1erejsan?dQ constantC,depending0j1onn,suchthat and beacubewithcenterandedgelength SOthatG(hQ)()<?.Thenthereisa G(hQ)()<? C(hQ)(x)一C(hQ)(x,)lcIIfllforall?dQ. Theproofoflemma3issimple,thereaderscanreferto Atlast,wedefinethegeneralizedareaintegralby )(('/I,?Jdydt wherer(x)=.((,t)?(R",(0,?)):J—YJ<t,t>0)andfor>0,wedefineother Littlewood—Paleyg—functionas ,)(厶州dydt. Thenwecanusethesimilarmethodtogetthesamerestfltsfors(f)andgA(,),thedetails areomited.Thereaderscanrefertof2]. 一 517一 \口 . 一.一 <一 References: 【1】WANGSL.SoreepropertiesofLittlewood—Paley'sg—functions[J].ScientiaSinieaA,1985, 28:252—262. 【2】KUR,TZDS.Littlewood—PaleyoperatorsOnBMO[J].Proc.Amer.Math.Soe.,1987,99(4) 657—666. 【3】DENGDG,HANYS.TheHpTheory[M].Beijing:BeijingUniversityPress,1992,16 —21? 广义Littlewood.Paley函数的BMO有界性 李文明 (河北师范大学数信学院,河北石家庄050016) 摘要:本文证明了BMO函数在广义Littlewood.Paley函数下的象或者几乎处处等于无 穷或者属于BMO. ...—— 518.--——