Sobolev spaces associated to the harmonic oscillat

arXiv:math/0608684v1[math.CO]28Aug2006Proc.IndianAcad.Sci.(Math.Sci.)Vol.116,No.3,August2003,pp.337–360.PrintedinIndiaSobolevspacesassociatedtotheharmonicoscillatorBBONGIOANNI1andJLTORREA21DepartamentodeMatem´atica,FacultaddeIngenier´ıaQ´ımica,UniversidadNacionaldelLitoral,andInstitutodeMatem´aticaAplicadadelLitoral,SantaFe(3000),Argentina2DepartamentodeMatem´atica,FacultaddeCiencias,UniversidadAut´onomadeMadrid,SpainE-mail:[e-mail];[e-mail]finetheHermite–SobolevspacesnaturallyassociatedtotheharmonicoscillatorH=−Δ+|x|2.Structuralproperties,relationswiththeclassicalSobolevspaces,boundednessofoperatorsandalmosteverywhereconvergenceofsolutionsoftheSchr¨odingerequationarealsoconsidered.Keywords.Hermiteoperator;potentialspaces;Riesztransforms.1.IntroductionWeconsiderthesecond-orderdifferentialoperatorH=−Δ+|x|2,x∈Rd.(1)Thisoperatorisself-adjointonthesetofinfinitelydifferentiablefunctionswithcompactsupportC∞c,anditcanbefactorizedasH=12d∑j=1AjA−j+A−jAj,(2)whereAj=∂∂xj+xjandA−j=A∗j=−∂∂xj+xj,1≤j≤d.InthelastfewyearsseveralauthorshavebeenconcernedwiththeharmonicanalysisassociatedtotheoperatorH(seeforinstance[5,10,12]).InthisanalysistheoperatorsAjplaytheroleofthepartialderivativeoperators∂/∂xjintheclassicalEuclideancase.HenceitseemsnaturaltostudythespacesoffunctionsinLp(Rd)whosederivativesalsobelongtoLp(Rd).Followingthisidea,weintroducetheHermite–SobolevspacesWk,p(Definition1).ThesespacesareBanachspacesandthesetoflinearcombinationsofHer-mitefunctionsisdenseinanyofthem(Proposition1).ThespacesWk,pwerepreviouslystudiedin[14]forp=2andin[7]forp6=2.OncewehavetheLaplacianH,itisalsonaturaltoconsiderthepotentialspacesLpa=H−a/2(Lp(Rd))(Definition3).Inotherwords,therangeoftheHermitefractional337338BBongioanniandJLTorreaintegraloperatorH−a/2inLp(Rd).InordertohaveasatisfactorydescriptionofthesepotentialspacesweneedasharpanalysisoftheoperatorH−a,fora0.SuchanalysisiscontainedinProposition2andLemma3.Itturnsoutthat,fork∈N,thespacesLpkandWk,pcoincide(seeTheorem4).TheproofofthistheoremusestheboundednessinLp(Rd)oftheRiesztransformsnaturallyassociatedtoH.TheseRiesztransformswereintroducedbyThangaveluin[12],andsomeoftheirboundednesspropertiescanbefoundin[5]and[10].Observethat,insomesense,Theorem4allowsustosaythatthespacesLpaarethespacesoffunctionsinLp(Rd)forwhichtheirderivativesoforderaalsobelongtoLp(Rd).OncewehaveasatisfactorydefinitionofHermite–Sobolev(orHermitepotential)spacesandhenceoffractionalderivatives,westudytheirrelationshipwiththecorre-spondingclassicalEuclideanspaces.WeshowinTheorem3thatalthoughtheHermite–SobolevspacescoincidelocallywiththeEuclideanSobolevspaces,theyareinfactdif-ferent.In§5.,weshowthattheHermite–RiesztransformsareboundedontheHermite–SobolevspaceswhiletheclassicalHilberttransformisnotboundedonthesespaces.FromthecarefulanalysisofthekernelofH−a,wealsoobtaincertaininequalitiesofPoincar´etypeforthederivativesAjinTheorem9.Finally,in§7.wegiveanapplicationtothealmosteverywhereconvergenceofthesolutionoftheSchr¨odingerequation(42)totheinitialdata.OurworkwasheavilyinspiredinthepaperbyThangavelu[14],wherethespacesL2aweredefined.Thesespaceswerealsoconsideredin[6].Aswesaidabove,inordertodevelopthiswork,somenontrivialestimatesoftheHer-mitefractionalintegraloperator(Definition2)wereneeded.However,itisnottheaimofthispapertomakeanexhaustivestudyofthisoperator.Hence,othernaturalquestionslikeweakandstrongboundednessintheextremepointsorBMO-typeboundednessoftheoperatorH−aareleftasideandtheywillbethemotivationofaforthcomingpaper.2.Hermite–SobolevspacesLetn∈N0=N∪{0}andconsidertheHermitefunctionofordern,hn(t)=(−1)n(2nn!π1/2)1/2Hn(t)e−t2/2,t∈R,whereHndenotestheHermitepolynomialofdegreen(see[12]).Givenamulti-indexα=(αj)dj=1∈Nd0,weconsidertheHermitefunction,hα,ashα(x)=d∏j=1hαj(xj),x=(x1,...,xd)∈Rd.ThesefunctionsareeigenvectorsoftheHermiteoperatorHdefinedin(1).InfactHhα=(2|α|+d)hα,where|α|=∑dj=1αj.Moreover,for1≤j≤d,Ajhα=p2αjhα−ej,A−jhα=q2(αj+1)hα+ej,whereejisthejthcoordinatevectorinNd0.TheoperatorsAjandA−jarecalledannihi-lationandcreationoperatorsrespectively.Sobolevspacesassociatedtotheharmonicoscillator339DEFINITION1.Givenp∈(1,∞)andk∈N,wedefinetheHermite–Sobolevspaceoforderk,denotedbyWk,p,asthesetoffunctionsf∈Lp(Rd)suchthatAj1···Ajmf∈Lp(Rd),1≤|j1|,...,|jm|≤d,1≤m≤k,withthenormkfkWk,p=∑1≤|j1|,...,|jm|≤d,1≤m≤kkAj1···Ajmfkp+kfkp.WewillshowthatthesetoffinitelinearcombinationsofHermitefunctions,denotedbyF,isdenseintheHermite–Sobolevspaces.Weshallneedthefollowinglemmas.Theirproofsmaybefoundin[10]and[12],respectively.Lemma1.Letm∈N0andf∈C∞c.ThereexistsaconstantCm,f0suchthat|hf,hαi|≤Cm,f(|α|+1)m,α∈Nd0.Lemma2.Asn→∞theHermitefunctionssatisfytheestimates(i)khnkp∼n12p−14,1≤p4,(ii)khnkp∼n−18log(n),p=4,(iii)khnkp∼n16p−112,4p≤∞.PROPOSITION1.Letpbeintherange1p∞andk∈N.ThesetWkpisaBanachspace.Moreover,thesetsFandC∞caredenseinWk,p.Proof.ToseethatW1,piscomplete,observethatif{fn}n≥1isaCauchysequenceinW1,p,then∂fn∂xjn≥1and{xjfn}n≥1,1≤j≤d,(3)areCauchysequencesinLp(Rd).IfwecallfthelimitinLp(Rd)of{fn}n