Boundedness on inhomogeneous Lipschitz spaces of f

arXiv:0809.3457v1[math.CT]19Sep2008BoundednessoninhomogeneousLipschitzspacesoffractionalintegrals,singularintegralsandhypersingularintegralsassociatedtonon-doublingmeasures.A.EduardoGatto1DePaulUniversityandCRMTomysonsAbstract:Inthecontextofafinitemeasuremetricspacewhosemeasuresatisfiesagrowthcondition,weprove”T1”typenecessaryandsufficientconditionsfortheboundednessoffractionalintegrals,singularintegrals,andhypersingularintegralsoninhomogeneousLipschitzspaces.Wealsoindicatehowtheresultscanbeextendedtothecaseofinfinitemeasure.FinallyweshowapplicationstoRealandComplexAnalysis.1ThispaperwassupportedbyasabbaticalleavefromDePaulUniversityandbyagrantfromtheMinisteriodeEducacionyCienciaofSpain,Ref.#SAB2006-0118foraquartersabbaticalvisittotheCentredeRecercaMatem`aticainBarcelona.Theauthorwantstoexpressadeepgratitudetotheseinstitutions,totheAnalysisgroupoftheUniversidadAut´onomadeBarcelona,inparticulartoXavierTolsafortheinvitationandtothedirectorandstaffoftheCRMfortheirpleasanthospitalityandexcellentresources.Keywords:non-doublingmeasures,metricspaces,Lipschitzspaces,fractionalintegrals,singularintegral,hypersingularintegrals.2000MSC:42B20,26B35,47B38,47G1011.Introduction.DefinitionsandStatementoftheTheoremsLet(X,d,μ)beafinitemeasuremetricspacewhosemeasureμsatisfiesan-dimensionalgrowthcondition,thatis,(X,d)isametricspaceandμisafiniteBorelmeasurethatsatisfiesthefollowingcondition:thereisn0andaconstantA0suchthatμ(Br)≤Arn,forallballsBrofradiusrandforallr0.Notethatthisconditionallowstheconsiderationofnon-doublingaswellasdoublingmeasures.Ourresultswillapplytofunctionsdefinedonthesupportofμ,ofcoursethesupportofμhastobewelldefined,wheresupp(μ)isthesmallestclosedsetFsuchthatforallBorelsetsE,E⊂Fc,μ(E)=0.Forexample,ifXisseparable,thenthesupportofμiswelldefined.FurthermoretoavoidanyconfusionwewillassumethatX=supp(μ)TheinhomogeneousLipschitz-H¨olderspacesoforderβ,0β≤1,willbedenotedΛβandconsistsofallboundedfunctionsfthatsatisfysupx6=y∈X|f(x)−f(y)|dβ(x,y)∞.ThespaceΛβisaBanachspacewiththenormkfkΛβ=supx∈X|f(x)|+supx6=y∈X|f(x)−f(y)|dβ(x,y).Itwillbeusefultohaveanotationforeachterminthenorm,letsup(f)=supx∈X|f(x)|,and|f|β=supx6=y∈X|f(x)−f(y)|dβ(x,y).Theresultsinthispaperhaveextensionstothecaseμ(X)=∞,buttheconstantsdependonthenormalizationoftheintegralsatinfinity,wewillindicatetheseextensionsafterthesectiononproofs.TheletterC,cwilldenoteconstantsnotnecessarilythesameateachocurrence.LetΩ=X×X\Δ,whereΔ={(x,y):x=y}.AfunctionLα(x,y):Ω→Cwillbecalledastandardfractionalintegralkerneloforderα,0α1,whenthereareconstantsB1andB2suchthat(L1)|Lα(x,y)|≤B1dn−α(x,y).(L2)|Lα(x1,y)−Lα(x2,y)|≤B2dγ(x1,x2)dn−α+γ(x1,y),forsomeγ,αγ≤1,and2d(x1,x2)≤d(x1,y).ThefractionalintegraloforderαofafunctionfinΛβisdefinedby:Lαf(x)=ZLα(x,y)f(y)dμ(y).NotethatinparticularLα(x,y)=1dn−α(x,y)isastandardfractionalkerneloforderα.Theorem1Let0αγ≤1,0β1,andα+β≤1when1norα+βnwhenn≤1.Thefollowingstatementsareequivalent:a)Lα1∈Λα+β.2b)Lα:Λβ→Λα+βisbounded.WedefinenowthesingularintegralkernelsthatwewillconsiderinTheorem2andTheorem3.AfunctionK(x,y):Ω→CwillbecalledastandardsingularintegralkernelwhenthereareconstantsC1,C2andanumberγ,0γ≤1,suchthat(S1)|K(x,y)|≤C1dn(x,y)(S2)|K(x1,y)−K(x2,y)|≤C2dγ(x1,x2)dn+γ(x1,y),for2d(x1,x2)≤d(x1,y)LetηbeafunctioninC1[0,∞)suchthat0≤η≤1,η(s)=0for0≤s≤1/2andη(s)=1for1≤s.LetKε(x,y)=η(d(x,y)ε)K(x,y),ε0whereK(x,y)isastandardsingularintegralkernel.WewilldenoteTεtheoperatorTεf(x)=RKε(x,y)f(y)dμ(y).Theorem2LetK(x,y)beastandardsingularintegralkernel.Let0βmin(n,γ).Thefollowingtwostatementsareequivalent:a)kTε1kΛβ≤C,forallǫ0.b)Tε:Λβ→ΛβareboundedandkTεkΛβ→Λβ≤C′,forallε0.OneofthenoveltiesinthisTheoremisthatthecancellationcondition(S3)forallx(seebelow)followsfrompartb).InTheorem3wewillconsiderPrincipalValueSingularIntegrals.WewilldenotebyLipβthespaceofclassesofmeasurablefunctionsfforwhichthereisag∈Λβsuchthatf=gexceptforasetEthatdependsonf,withμ(E)=0.ThenormoffinLipβisdefinedaskfkLipβ=kfk∞+|f|β,where|f|β=supx6=y∈X|g(x)−g(y)|dβ(x,y)=supx6=y∈X−E|f(x)−f(y)|dβ(x,y).Wealsoneedtoaddthefollowingtwoconditionsonthekernel:(S3)Rr1d(x,y)r2K(x,y)dμ(y)≤C3forall0r1r2∞,μ−a.einx.(S4)limε→0Rεd(x,y)1K(x,y)dμ(y)existsμ−a.einx.Theprincipalvaluesingularintegralofafunctionf∈LipβisdefinedbyKf(x)=limε→0Zεd(x,y)K(x,y)f(y)dμ(y)Theorem3LetK(x,y)beastandardsingularintegralkernelthatinadditionsat-isfies(S3)and(S4).Let0βmin(n,γ)andf∈Lipβ.ThenKf(x)iswelldefinedμ−a.e.andthefollowingtwostatementsareequivalent:3a)K1∈Lipβb)K:Lipβ→LipβisboundedAfunctionDα(x,y):Ω→Cwillbecalledastandardhypersingularkerneloforderα,0α1,whenthereareconstantsE1andE2suchthat:(D1)|Dα(x,y)|≤E1dn+α(x,y),(D2)|Dα(x1,y)−Dα(x2,y)|≤E2dγ(x1,x2)dn+α+γ(x1,y),forsomeγ,0γ≤1,and2d(x1,x2)≤d(x1,y).Thehypersingularintegraloforderαofafunctionf∈Λβαβ≤1isdefinedby:Dαf(x)=ZDα(x,y)[f(y)−f(x)]dμ(y)NotethatinparticularDα(x,y)=1dn+α(x,y)isastandardhypersingularker-neloforderαwhenX=RnandμistheLebesguemeasure,andwehaveR1dn+α(x,y)[f(y)−f(x)]dy=cα(Δα2f)(x)forfsufficientlysmoothand0α2.[S]Theorem4Let0αβ≤1andβ−αn.ThenDα:Λβ→Λβ−αisbounded.NotethatDα1=0byd