Regularity theory for the fractional harmonic osci

JournalofFunctionalAnalysis260(2011)3097–3131✩PabloRaúlStingaa,JoséLuisTorreab,∗aDepartamentodeMatemáticas,FacultaddeCiencias,UniversidadAutónomadeMadrid,28049Madrid,SpainbDepartamentodeMatemáticasandICMAT-CSIC-UAM-UCM-UC3M,FacultaddeCiencias,UniversidadAutónomadeMadrid,28049Madrid,SpainReceived28April2010;accepted1February2011Availableonline12February2011CommunicatedbyGillesGodefroyAbstractInthispaperwedevelopthetheoryofSchauderestimatesforthefractionalharmonicoscillatorHσ=(−+|x|2)σ,0σ1.Moreprecisely,anewclassofsmoothfunctionsCk,αHisdefined,inwhichwestudytheactionofHσ.InfactthesespacesarethoseadaptedtotheoperatorH,hencethesuitedonesforthistypeofregularityestimates.Inordertoproveourresults,ananalysisoftheinteractionoftheHermite–RiesztransformswiththeHölderspacesCk,αHisneeded,thatwebelieveofindependentinterest.©2011ElsevierInc.Allrightsreserved.Keywords:Fractionalharmonicoscillator;Schauderestimate;Fractionalintegral;Riesztransform1.IntroductionForagivenpartialdifferentialoperatorL,theanalysisofitsregularitypropertieswithrespecttoHölderclassesisoneofthetoolsemployedinthetheorytoproveimportantfactsaboutpartialdifferentialequations.Indeed,beingabitimprecise,itiswellknownthatiffisaHöldercon-tinuousfunctionwithexponentα,thentheequation−u=fhasauniquesolutionu,whosesecondorderderivativesbelongtoCα,anduC2,αiscontrolledbyfCα.ThisresultwasfirstappliedtoobtainclassicalsolutionsofsecondorderellipticequationsoftheformLu=f(seeforinstance[7,Chapter6]).Recently,andmotivatedbytheobstacleproblemforthefractional✩ResearchsupportedbyMinisteriodeCienciaeInnovacióndeEspañaunderprojectMTM2008-06621-C02-01.*Correspondingauthor.E-mailaddresses:[e-mail](P.R.Stinga),[e-mail](J.L.Torrea).0022-1236/$–seefrontmatter©2011ElsevierInc.Allrightsreserved.doi:10.1016/j.jfa.2011.02.0033098P.R.Stinga,J.L.Torrea/JournalofFunctionalAnalysis260(2011)3097–3131Laplacian,L.Silvestreprovedin[11](seealsohisthesis[10])theregularitypropertiesfortheoperator(−)σ,0σ1,whenactingonHölderspaces.Formoreapplicationssee[5]and[6].LetHbethemostbasicSchrödingeroperatorinRn,n1,theharmonicoscillator:H=−+|x|2.ThefractionalpowersHσ,0σ1,wereintroducedin[13].TheaimofthispaperistoproveregularityestimatesinHölderclassesforthefractionalharmonicoscillatorHσ.Forthispurpose,wedefinenewHölderspacesCk,αH,differentthantheclassicalHölderspacesCk,α,inwhichthesmoothnesspropertiesofHσareanalyzed,seeDefinition1.1andTheoremA.TheclassesCk,αHarethenaturalspacesassociatedtoH.Thisbecomesevident,forinstance,inthefactthattheHermite–Riesztransformshavetheexpectedbehavior:theypreservethem,seeTheorem4.1.Alsothefractionalintegralsproduceakindof“inversefractionalderivative”processwhenactinginCk,αH,seeTheoremB.Ourestimates,togetherwithHarnack’sinequalityforHσprovedin[13],arethebasicregu-larityestimatesoneexpectstogetforthefractionalpowersofasecondorderoperator.Moreover,wefoundtherightspacesforwhichSchauderestimatesareappropriated.Weexpecttoobtainthecorrectregularityestimatesfornonlinearproblemsrelatedtothefractionalharmonicoscillatorinthesespaces.Applicationswillappearelsewhere.LetusintroducethedefinitionofHσ.ForafunctionfinSchwartz’sclassSand0σ1,thefractionalharmonicoscillatorHσisgivenbytheclassicalformulaHσf(x)=1Γ(−σ)∞0e−tHf(x)−f(x)dtt1+σ,(1.1)wherev(x,t)=e−tHf(x)isthesolutionoftheheat-diffusionequation∂tv+Hv=0inRn×(0,∞),withinitialdatumv(x,0)=f(x)onRn.In[13]itisshownthatHσf(x)=Rnf(x)−f(z)Fσ(x,z)dz+f(x)Bσ(x),x∈Rn,f∈S,(1.2)whereFσ(x,z)=1−Γ(−σ)∞0Gt(x,z)dtt1+σ,Bσ(x)=1Γ(−σ)∞0RnGt(x,z)dz−1dtt1+σ,(1.3)andGt(x,z)isthekerneloftheheat-diffusionsemigroupgeneratedbyH,see(3.1).NextwedefinetheHölderspacesinwhichtheregularitypropertiesoftheoperatorswillbeconsidered.P.R.Stinga,J.L.Torrea/JournalofFunctionalAnalysis260(2011)3097–31313099Definition1.1.Let0α1.Acontinuousfunctionu:Rn→RbelongstotheHermite–HölderspaceC0,αHassociatedtoH,ifthereexistsaconstantC,dependingonlyonuandα,suchthatu(x1)−u(x2)C|x1−x2|α,andu(x)C(1+|x|)α,forallx1,x2,x∈Rn.Withthenotation[u]C0,α=supx1,x2∈Rnx1=x2|u(x1)−u(x2)||x1−x2|α,and[u]Mα=supx∈Rn1+|x|αu(x),wedefinethenorminthespacesC0,αHtobeuC0,αH=[u]C0,α+[u]Mα.WhenworkingwiththeharmonicoscillatorHsomespecialfirstorderpartialdifferentialoperatorsareconsidered,see(3.3),whicharethenaturalderivatives.ThentheclassesCk,αHcanbedefinedinausualway,seeDefinition3.1.Wepresentnowourfirstmainresult.TheoremA.Letα∈(0,1]andσ∈(0,1).(A1)Letu∈C0,αHand2σα.ThenHσu∈C0,α−2σHandHσuC0,α−2σHCuC0,αH.(A2)Letu∈C1,αHand2σα.ThenHσu∈C1,α−2σHandHσuC1,α−2σHCuC1,αH.(A3)Letu∈C1,αHand2σα,withα−2σ+1=0.ThenHσu∈C0,α−2σ+1HandHσuC0,α−2σ+1HCuC1,αH.(A4)Letu∈Ck,αHandassumethatk+α−2σisnotaninteger.ThenHσu∈Cl,βHwherelistheintegerpartofk+α−2σandβ=k+α−2σ−l.ThelasttheoremcanbeinterpretedassayingthattheHölderspacesCk,αHarethereasonableclassesinordertoobtainSchaudertypeestimatesforHσ.Indeed,ifwedefinethenegativepowersofH,i.e.thefractionalintegraloperatorsH−σf(x)=1Γ(σ)∞0e−tHf(x)dtt1−σ,0