Wavelets, Multiplier spaces and application to Schrödinger type operators with non-smooth potentials PDF
Preview Wavelets, Multiplier spaces and application to Schrödinger type operators with non-smooth potentials
WAVELETS, MULTIPLIER SPACES AND APPLICATION TO SCHRO¨DINGER TYPE OPERATORS WITH NON-SMOOTH POTENTIALS 3 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU 1 0 2 n Abstract. Inthispaper,weemployMeyerwaveletstocharacterizemultiplierspacesbe- a J tweenSobolevspaceswithoutusingcapacity. Further,weintroducelogarithmicMorrey 4 spacesMrt,,τp(Rn)toestablishtheinclusionrelationbetweenMorreyspacesandmultiplier spaces. Bywaveletcharacterization andfractalskills,weconstructacounterexample to ] P showthatthescopeoftheindexτofMrt,,τp(Rn)issharp. Asanapplication,weconsidera A Schro¨dingertypeoperatorwithpotentialsinMt,τ(Rn). r,p . h t 1. Introduction a m AfunctiongiscalledamultiplierfromHt+r,p(Rn)toHt,p(Rn)ifforeveryfunction f [ ∈ Ht+r,p(Rn),theproduct fg Ht,p(Rn).WedenotebyXt (Rn)theclassofallsuchfunctions ∈ r,p 1 g.Asusefultools,multipliersonthespacesofdifferentialfunctionsareappliedtothestudy v 6 of various problems in harmonic analysis and differential equations. For example, the 9 coefficientsofadifferentialoperatorcanbeseenasmultipliers.Forafunctionubelonging 6 0 tosomeBanachspace,M.Cannoneremindedusthatthenonlineartermu2canberegarded . 1 astheproductofafunctionuinthisBanachspaceandamultiplieru. M.Cannonemade 0 3 many contributions on nonlinear problems. See [1, 2, 3, 8]. For more information on 1 bothmultiplierspacesandPDE,wereferthereaderstoV.Maz’yaandT.Shaposhnikova’s : v celebratedmonograph[10]andtheirrecentwork. i X In[10],V.Maz’yaandT.Shaposhnikovagavemanycharacterizationsofdifferentkinds r a ofmultiplierspaces. Forexample,theyobtainedthatfort 0,r > 0,1 < p < n/(r+t), ≥ themultiplierspacesXt (Rn)canbecharacterizedbycapacityonarbitrarycompactsets. r,p ThemultiplierspacesXt (Rn)aredefinedasfollows. r,p Definition1.1. ([10])Fix1< p< andr,t 0. ThemultiplierspaceXt (Rn)isdefined ∞ ≥ r,p asthesetofallthefunctions f(x)suchthat kfkXrt,p(Rn) := sup kfgkHt,p(Rn) <∞. kgkHt+r,p(Rn)≤1 2000MathematicsSubjectClassification. Primary42B20;76D03;42B35;46E30. Keywordsandphrases. Daubechieswavelets,multiplierspaces,Sobolevspaces,logarithmicMorreyspaces. TheresearchissupportedbyNSFCNo. 11171203; NewTeachers’FundforDoctorStations, Ministryof Education20114402120003;andGuangdongNaturalScienceFoundationS2011040004131. 1 2 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU Foracompactsete Rn,thecapacitycap(e,Ht,p)oneisdefinedby ⊂ cap(e,Ht,p)=inf u p : u ,u 1one , k kHt,p(Rn) ∈S ≥ n o where istheSchwartzclassofrapidlydecreasingsmoothfunctionsonRn. S Lemma1.2. ([10])Givenr>0andt 0. ≥ (i)For1< p<n/(r+t), f Xt (Rn)ifandonlyif ∈ r,p ( ∆)t/2f f sup k − kLp(e) + k kLp(e) < . e Rn (cap(e,Ht+r,p))1/p (cap(e,Hr,p))1/p! ∞ ⊂ (ii)For1 < p < n/randanycubeQwithlengthlessthan1,thecapacitycap(Q,Hr,p) islessthanCQ1 pr/n. − | | Our motivation is based upon the following consideration. For complicated compact sets,itisverydifficulttocomputethecapacity.Themainaimofthispaperistogivesome wavelet characterizationsand introducesome sufficient conditionswhich can be verified easily. Precisely,forr > 0,t 0andt+r < 1 < p < n/(r+t),wewillgiveacharacter- ≥ ization of the multipliersfrom Ht+r,p(Rn) and Hr,p(Rn) by Meyer wavelets withoutusing capacity.SeeTheorems3.3. Alsoourmethodcanbeappliedtostudytherelationbetween multiplierspacesandMorreyspaces. Todealwiththecaset>0,wehavetointroducethe almostlocaloperatorTt. See 2. § Lemma 1.2 implies that the multiplier space Xt (Rn) is a subspace of Morrey space r,p Mt (Rn). It is natural to ask if the reverse inclusion relation holds. Unfortunately, for r,p t=0,theimbeddingXt (Rn) Mt (Rn)isnotanisomorphism.In[8],P.G.Lemarie´gave r,p ⊂ r,p a counterexampletoshowthatwhenn 2r is aninteger, X0 (Rn) , M0 (Rn). Recently, − r,2 r,2 P. G. Lemarie´ [9] and Yang-Zhu [23] constructed some counterexamples for t = 0 and 1< p<n/r. Fort >0,wehavetoconsidertheactionofthedifferentiation,sowecannotconstruct counterexamplelikethecaset = 0in[23]. OurcounterexampleinTheorem5.4depends onourwaveletcharacterization,Theorem5.3andfractalskills. Fromthiscounterexample, we can see that the productof f Mt (Rn) and g Ht+r,p(Rn) may producea blow up ∈ r,p ∈ phenomenonoflogarithmictypeonfractalsetswithHausdorffdimensionn p(r+t). To − eliminate this defect, we introduce a logarithmic type Morrey space Mt,τ(Rn) and prove r,p thatforτ>1/p, ′ Mt,τ(Rn) Xt (Rn) Mt (Rn)= Mt,0(Rn), r,p ⊂ r,p ⊂ r,p r,p wherer>0, t 0and1< p<n/(r+t). See 4. ≥ § Itshouldbepointedoutthat,intheaboveinclusionrelation,thescopeofτis(1/p, ), ′ ∞ where p is the conjugate number of p. In 5, our counterexample implies that, for ′ § 0 < τ 1/p , there exists some function f Mt,τ(Rn), but f < Xt (Rn). See 5 for ≤ ′ ∈ r,p r,p § WAVELETS,MULTIPLIERSPACESANDSCHRO¨DINGERTYPEOPERATORS 3 the details. Theorems 5.3 and 5.4 illustrate the difference between Morrey spaces and multiplierspaces. Another motivation is that the results about multipliers on Sobolev spaces can be ap- plied to the study on partial differential equations. For example, in [11], V. Maz’ya and I.E. Verbitskyconsideredthe multipliersfrom H1,2(Rn) to H 1,2(Rn). Fora Schro¨dinger − operator L = I ∆+V, they got many sufficient and necessary conditions such that V − isa multiplierfrom H1,2(Rn) to H 1,2(Rn). Formoreinformation,we referthereadersto − [8,10,11,12]andthereferencestherein. Asanapplicationofourresults,weconsiderthesolutioninSobolevspacesHt+r,p(Rn) totheequation: (1.1) (I+( ∆)r/2+V)f = g, − whereg Ht,p(Rn)andV Mt,τ(Rn)withr>0,t 0,1< p<n/(r+t),τ>1/p. IfV is r,p ′ ∈ ∈ ≥ afunctionofHo¨lderclass,oneusualmethodtodealwithequation(1.1)istheboundedness ofCaldero´n-Zygmundoperators.AsafunctioninthelogarithmicMorreyspacesMt,τ(Rn), r,p V maybe nota L function. In 6,byTheorem4.8, we provethatforV(x) Mt,τ(Rn), ∞ r,p § ∈ theaboveequation(1.1)hasanuniquesolutionintheSobolevspaceHt+r,p(Rn). Inthispaper,weusefourtoolsinanalysis. Oneisthemulti-resolutionanalysisintro- ducedbyY.MeyerandS.Mallatin1990s. TheotheristhealmostlocaloperatorTt. See 2.Bytheprojectionoperatorsgeneratedfrommulti-resolutionanalysis,S.Dobynski(cf. § [4] ) obtained a decompositionof the productof two functions. In order to adapt to our needs,wemakesomemodificationtoDobynski’sdecomposition.Thethirdmainskillsare tousecombinationatomsandtointroducesomedifferentialmethods.Theforthmainskill istochoosespecialfunctionssuchthattheirwaveletcoefficientsrestrictedonsomefractal sets. See 4 and 5. § § Our paper is organizedas follows. In 2, we state some notationsand known results § which will be used throughout this paper. In 3, we give a wavelet characterization of § the multiplier spaces Xt (Rn). In 4, we introducea class of logarithmicMorreyspaces r,p § Mt,τ(Rn) and get a very simple sufficient condition of Xt (Rn). In 5, for Mt,τ(Rn), we r,p r,p § r,p constructacounterexampletoprovethesharpnessofthescopeoftheindexτobtainedin 4.Inthelastsection,weconsideranapplicationtoPDEproblem. § 2. Somepreliminaries Inthissection,westatesomenotations,knowledgeandpreliminarylemmaswhichwill be used in the sequel. Firstly, we recall some background knowledge of wavelets and multi-resolutionanalysis. Wewilladoptreal-valuedtensorproductwaveletstostudythemultiplierspacesinthis paper. Let E = 0,1 n 0 . For ε = 0 (respectively, ε E ), we assume that Φε(x) is n n { } \{ } ∈ 4 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU a scaling function(respectively,wavelet). In the proof,we use onlyMeyerwavelets and regularDaubechieswavelets. We say a Daubechieswavelet is regularif it has sufficient vanishingmomentuntilordermandΦε(x) Cm([ 2M,2M]),wheretheregularityexponent ∈ 0 − m is large enough and M is determined by m, see [13, 18] for more details. For any ε 0,1 n, j Nandk Zn,wedenoteΦε (x)=2jn/2Φε(2jx k). Inadditionwedefine ∈{ } ∈ ∈ j,k − Λ = (ε, j,k):ε 0,1 n, j N,k Znandε,0, if j>0 . n ∈{ } ∈ ∈ n o Forfixedtempereddistribution f, if we use waveletswhichissufficientregular,thenwe candefine fε = f,Φε . Andthewaveletrepresentation f = fε Φε holdsinthe j,k j,k j,k j,k senseofdistributDion. E (ε,j,Pk)∈Λn Let V1, j Z be an orthogonal multi-resolution in L2(R) with the scaling function j ∈ Φ0(x).Dn enotebyoW1theorthogonalcomplementspaceofV1inV1 ,thatis,W1 =V1 j j j+1 j j+1⊖ V1. Let Φ1(x k), k Z be an orthogonalbasis in W1. Forε = (ε , ,ε ) 0,1 n, j − ∈ 0 1 ··· n ∈ { } n n o denote Φε(x) = i=1Φεi(xi). For Vj = (f(x)=k Zn fj0,kΦ0j,k(x), where{fj0,k}k∈Zn ∈l2) and Q P∈ Wj =(f(x)=ε∈EPn,k∈Zn fjε,kΦεj,k(x), where{fjε,k}ε∈En,k∈Zn ∈l2),wehave Lemma2.1. V , j Z isanorthogonalmulti-resolutionwiththescalingfunctionΦ0(x). j { ∈ } W is the orthogonalcomplementspace of V in V , that is, W = V V . Further j j j+1 j j+1 j ⊖ Φε ,(ε, j,k) Λ isanorthogonalbasisinV W = L2(Rn). j,k ∈ n 0 j n o Lj 0 ≥ Denoteby P and Q theprojectionoperatorsfrom L2(Rn)toV andW , respectively. j j j j ByLemma2.1,S.Dobynskigotadecompositionoftheproductoftwofunctions f andg, whichissimilartoBony’sparaproduct(see[4]). Denote Λ˜ = (ε,ε, j,k,k ), ε,ε 0,1 n 0 , j 0,k,k Zn,(ε,k),(ε,k ) . n ′ ′ ′ ′ ′ ′ ∈{ } \{ } ≥ ∈ n o By the projection operators P and Q , we divide the product of f(x) and g(x) into the j j followingterms. f(x)g(x) = P (f)P (g)+ P (f)Q (g)+ Q (f)P (g) 0 0 j j j j Xj 0 Xj 0 ≥ ≥ + fε gε′ Φε (x)Φε′ (x)+ fε gε Φε (x) 2. j,k j,k j,k j,k j,k j,k j,k XΛ˜n ′ ′ ΛXn,ε,0 (cid:16) (cid:17) To facilitate our use, we make a modification to the above decomposition and use spe- cialwaveletsfordifferentcases. Let N bea positiveinteger. We decomposethe product WAVELETS,MULTIPLIERSPACESANDSCHRO¨DINGERTYPEOPERATORS 5 f(x)g(x)as ∞ fg= P (f)P (g) P (f)P (g) +P (f)P (g) j+1 j+1 j j 0 0 − Xj=1h i (2.1) ∞ = Q (f)Q (g)+P (f)Q (g)+Q (f)P (g) +P (f)P (g) j j j j j j N N Xj=Nh i andtheterm ∞ Q (f)P (g)canbedecomposedas j j j=N P N ∞ ∞ ∞ Q (f)P (g)= Q (f) Q (g)+ Q (f)P (g) j j j j t j j N Xj=N Xj=N Xt=1 − Xj=N − (2.2) N ∞ ∞ = Q (f)Q (g)+ Q (f)P (g). j+t j j+N j Xj=0Xt=1 Xj=0 n For any j N and k = (k ,k ,...,k ) Zn, let Q = [2 jk ,2 j(k + 1)] and 1 2 n j,k − s − s ∈ ∈ s=1 denote by Ω the set of all dyadic cubes Q . For arbitrary seQt Q, we denote by Q˜ the j,k 2M+2 multipleofQ. Finally,letχ(x)bethecharacteristicfunctionoftheunitcubeQ and 0 − χ˜ bethecharacteristicfunctionofQ˜ . 0 In1970s,H.TriebelintroducedTriebel-LizorkinspacesFr,q(Rn)([17]). Manyfunction p spacescanbeseenasthespecialcasesforFr,q(Rn). Forexample,Fr,2(Rn)isthefractional p 1 Hardy space. For 1 < p < , Fr,2(Rn) are the Sobolev spaces Hr,p(Rn). For p = , p ∞ ∞ F−r,2(Rn)isthefractionalBMOspace BMOr(Rn)whichisdefinedby BMOr(Rn) := (I ∞ − ∆) r/2BMO(Rn), where I is the unitoperator,∆ is the Laplaceoperator. Here BMO(Rn) − denotesthenon-homogeneousboundedmeanoscillationspacewhichisdefinedastheset ofthefunctionssuchthat 1 sup f =sup f(x)dx C |Q|=1| Q| Q |Q|(cid:12)(cid:12)(cid:12)(cid:12)ZQ (cid:12)(cid:12)(cid:12)(cid:12)≤ (cid:12) (cid:12) (cid:12) (cid:12) and 1 sup f(x) f 2dx< . Q Q Z | − | ∞ |Q|≤1| | Q For1 ≤ p < ∞andr ∈ R,itiswellknownthat Frp,2(Rn) ′ = F−p′r,2(Rn). Thefollowing lemma gives a characterization of Fr,2(Rn) by Me(cid:16)yer wave(cid:17)lets and regular Daubechies p wavelets. Fortheproof,wereferthereadersto[20,24]. 6 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU Lemma 2.2. (i) For 1 p < and r < m, using Meyer wavelets or m-regular ≤ ∞ | | Daubechieswavelets,wehavethefollowingequivalentcharacterizations, g(x)= gε Φε (x) Fr,2(Rn) j,k j,k ∈ p (ε,Xj,k)∈Λn 1/2 ⇐⇒ (cid:13)(cid:13) 22j(r+n/2)|gεj,k|2χ(2j−k) (cid:13)(cid:13) <∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(ε,Xj,k)∈Λn 1/2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Lp (cid:13) (cid:13) ⇐⇒ (cid:13)(cid:13) 22j(r+n/2)|gεj,k|2χ˜(2j−k) (cid:13)(cid:13) <∞. (ii)Given r <m.g(x)= (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(ε,Xj,k)g∈εΛnΦε (x) Fr,2(Rn)ifandonl(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)yLpifthereexists1< p< suchthatf|or| anyQ Ω, (ε,j,Pk)∈Λn j,k j,k ∈ ∞ ∞ ∈ 1/2 (cid:13)(cid:13) 22j(r+n/2)|gεj,k|2χ(2j−k) (cid:13)(cid:13) ≤C|Q|1/p. (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)ε∈EnX,Qj,k⊂Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Lp Thewaveletcha(cid:13)racterizationsoffunctionspaceshaveb(cid:13)eenstudiedbymanyauthors.In Chapters5and6of[13],Y.Meyerestablishedwaveletcharacterizationsformanyfunction spaces. In [22], Q. Yang, Z. Cheng and L. Peng considered wavelet characterization of Lorentz type Triebel-Lizorkin spaces and Lorentz type Besov spaces. In [20], Q. Yang introducedthe wavelet definitionof Besov type Morreyspaces. W. Yuan, W. Sickel and D.YangconsideredtheatomicdecompositionforBesovtypeMorreyspacesandTriebel- LizorkintypeMorreyspacesin[24]. MorreyspacesMt (Rn)wereintroducedbyMorreyin1938andplayanimportantrole r,p in the research of partial differential equations. In 2003, Wu and Xie [19] proved that generalized Morrey spaces are also generalization of Q-type spaces. In recent 20 years, Q-typespacesarestudiedextensively(cf[6,15,20,24]). Let f bethemeanvalueof(I ∆)t/2f onacubeQ, t,Q − 1 f = (I ∆)t/2f(x)dx. t,Q Q Z − | | Q TheMorreyspacesMt (Rn)aredefinedasfollows. r,p Definition2.3. For1 p < andr, t 0,theMorreyspace Mt (Rn)isdefinedasthe ≤ ∞ ≥ r,p setofthefunctions f suchthat sup f Cand t,Q | |≤ Q=1 | | (I ∆)t/2f(x) f pdx CQ1 p(r+t)/n, t,Q − Z − − ≤ | | Q(cid:12) (cid:12) (cid:12) (cid:12) whereQisanycubeinRn(cid:12)with Q 1. (cid:12) | |≤ In [15, 24], the authorsprovedthatMorreyspaces Mt (Rn) can be also characterized r,p bywavelets. Westateitasthefollowingtheoremandreferto[24]fortheproof. WAVELETS,MULTIPLIERSPACESANDSCHRO¨DINGERTYPEOPERATORS 7 Theorem2.4. Givent R,1< p< and0 p(r+t)<n. ∈ ∞ ≤ f(x)= fε Φε (x) Mt (Rn) j,k j,k ∈ r,p (ε,Xj,k)∈Λn ifandonlyifforanyQ Ωwith Q 1, ∈ | |≤ p/2 2j(n+2t) fε 2χ(2jx k) dx CQ1 p(r+t)/n. By LemmasZ1Q.2εa∈nEdnX,Q2j.,k2⊂,Qthe mul|tipj,kl|ierspac−es Xt (Rn)≤are|als|o−subspacesof Morrrey r,p spacesMt (Rn). r,p Lemma 2.5. Given r > 0, t 0 and 1 < p < n/(r+t). If f Xt (Rn), then f(x) ≥ ∈ r,p ∈ Mt (Rn). r,p Now we give two lemmas about the fractional BMO spaces BMOr(Rn). In the first lemma,weprovethatMorreyspacesMt (Rn)aresubspacesofBMOr(Rn). r,p Lemma2.6. Forr>0, t 0and1< p<n/(r+t), Mt (Rn) BMOr(Rn). ≥ r,p ⊂ Proof. ForanydyadiccubeQ,wehave p/2 2jn 2jr fε 2χ(2jx k) dx Z ε∈EnX,Qj,k⊂Q − | j,k| − p/2 Qp(r+t)/n 2j(n+2t) fε 2χ(2jx k) dx ≤ C| Q| p(r+t)/ZnQε1∈EpnX(r,Q+tj,)k/⊂nQ | j,k| − − ≤ | | | | CQ. ≤ | | (cid:3) Lemma 2.7. Suppose r > 0 and f = fε Φε (x) BMOr(Rn). The wavelet coefficientsof f satisfy (ε,j,Pk)∈Λn j,k j,k ∈ fε C2(r n/2)j, ε 0,1 n, j N, k Zn. | j,k|≤ − ∀ ∈{ } ∈ ∈ Proof. Take j Nandk Zn. Weconsidertwocasesε E andε=0separately. n ∈ ∈ ∈ (i)Foranyε E ,bythedefinitionofBMOr(Rn),weget n ∈ 2jn 2jr fε 2χ(2jx k) p/2dx C2 jn. Z − | j,k| − ≤ − (cid:16) (cid:17) Itiseasytoseethat fε C2j(r n/2). | j,k|≤ − (ii)Forε=0, f0 = fε′ Φε′ ,Φ0 = fε′ Φε′ ,Φ0 . j,k *(ε′,jX′,k′)∈Λn j′,k′ j′,k′ j,k+ *Xj′<j j′,k′ j′,k′ j,k+ 8 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU Because(cid:12)(cid:12)(cid:12)jP′<jfjε′′,k′Φεj′′,k′(x)(cid:12)(cid:12)(cid:12)≤C2rj,wehave|fj0,k|≤CD2rj,|Φ0j,k(x)|E≤C2j(r−n/2). (cid:3) (cid:12) (cid:12) LetΨ1(cid:12)(cid:12)andΨ2betwofu(cid:12)(cid:12)nctionsinCµ([ 2M+1,2M+1]n)withvanishingmoments xαΨi(x)dx= 0 − 0,where α µandi=1,2.Denote R | |≤ a = Ψ1 , Ψ2 . j,k,j′,k′ j,k j′,k′ D E ThefollowinglemmacanbefoundinChapter8of[13]orChapter6of[20]. Lemma 2.8. Given µ m. For s < µ, the coefficients a satisfy the following j,k,j,k | | ≤ | | ′ ′ condition: (2.3) |aj,k,j′,k′|≤C2−|j−j′|(n/2+s) 2−j+2−2j′−+j+|k22−−jj′−k′2−j′|!n+s. BywaveletcharacterizationofHr,p(Rn),thecontinuityofCaldero´n-Zygmundoperators onHr,p(Rn)isequivalenttothefollowinglemma. Wereferthereadersto[13,14,20]for theproof. Lemma 2.9. Suppose s > r and g(x) = gε Φε (x) Hr,p(Rn). Let g˜ε = | | (ε,j,Pk)∈Λn j,k j,k ∈ j,k aε,ε′ gε . Ifthecoefficientsaε,ε′ satisfythecondition(2.3),thenwehave j,k,j,k j,k j,k,j,k (ε,j,Pk)∈Λn ′ ′ ′ ′ ′ ′ p/2 p/2 2j(n+2r)g˜ε 2χ(2j k) dx C 2j(n+2r)gε 2χ(2j k) dx. Z | j,k| − ≤ Z | j,k| − (ε,Xj,k)∈Λn (ε,Xj,k)∈Λn We say thatT is a localoperatorif there existssome constantC > 1 suchthatforall x Rn and r > 0, T maps a distribution with the support B(x,r) to another distribution ∈ supportedontheballB(x,Cr).Ift/2isnotanon-negativeinteger,theoperator(I ∆)t/2is − notalocaloperator. Nowweusewaveletstoconstructsomespecialfractionaldifferential operatorsTt, whichare almost localoperatorsandwill be used in the proofof ourmain result. Definition2.10. Fort 0andh(x) = hε Φε (x),wedefineanoperatorTt corre- spondingtothekernelK≥t(x,y)= (ε,2j,Pk)∈jtΛΦnε j(,kx)Φj,kε (y)as − j,k j,k (ε,j,Pk)∈Λn Tth(x)= 2 jthε Φε (x). − j,k j,k (ε,Xj,k)∈Λn ItiseasytoprovethatT0istheidentityoperatorand Tth = h for1< p< . Lp H t,p k k k k − ∞ Furthermore,wehave Lemma2.11. Supposet 0. Forany Q Ωand x Q , 2j(n/2 t)h0 CMTth(x), ≥ j,k ∈ ∈ j,k − | j,k| ≤ whereMistheHardy-Littlewoodmaximaloperator. WAVELETS,MULTIPLIERSPACESANDSCHRO¨DINGERTYPEOPERATORS 9 Proof. Ift=0,theproofwasgivenbyMeyer[13]. Nowweconsiderthecaset>0. Since T tΦ0(x)= K t(x,y)Φ0(y)dy,itiseasytoverifythat − − R T tΦ0(x) C(1+ x) n t. − − − | |≤ | | Bythefactthatt>0,wehave 2j(n/2 t)h0 =2j(n/2 t) Tth(x), T tΦ0 (x) =2jn/2 Tth(x), (T tΦ0) (x) . − j,k − − j,k − j,k D E D E Hencewecanget 2j(n/2 t)h0 = 2jn/2 Tth(x), (T tΦ0) (x) | − j,k| − j,k (cid:12)(cid:12)(cid:12)D E(cid:12)(cid:12)(cid:12)dx C2jn/(cid:12)2 Tth(x)2jn/2 (cid:12) ≤ Z | | (1+ 2jx k)n+t | − | ∞ dx C2jn Tth(x)dx+ Tth(x) ≤ Z|2jx−k|≤1| | Xl=1 Z2l−1<≤|2jx−k|≤2l| |(1+|2jx−k|)n+t C2jn 2 jnM(Tth)(x)+ ∞ 2 ltM(Tth)(x)2 jn ≤ − − − CM(Tt)h(x). Xl=1 ≤ ThiscompletestheproofofLemma2.11. (cid:3) Intherestofthissection, wegiveadecompositionofSobolevspacesassociatedwith combinationatoms. For r <mandg(x)= gε Φε (x),denote | | (ε,j,Pk)∈Λn j,k j,k 1/2 S g(x)= 2j(2r+n)gε 2χ(2jx k) r | j,k| − (ε,Xj,k)∈Λn andfort=0,denotealsoSg(x)=S g(x). 0 Definition2.12. Givenr R, λ>0. ForarbitrarymeasurablesetE,wesaythatg(x)isa ∈ (r,λ,E) combinationatom,if supp(S g) E andS g(x) λ. IfE isadyadiccube,then r r − ⊂ ≤ g(x)iscalleda(r,λ,E) atom. − In[21],Q.YangintroducedthecombinationatomdecompositionofLebesguespaces. Inthispaper,weneedasimilarresultforSobolevspaces. Theorem2.13. If1 < p < , r < mand g 1, thereexistsaseriesof(r,2v,E )- Hr,p v ∞ | | k k ≤ combinationatomsg (x)suchthat 2pvE C. v v v N | |≤ P∈ Proof. Denote 1/2 S˜ g(x)= 2j(2r+n)gε 2χ˜(2j k) . r | j,k| − (ε,Xj,k)∈Λn 10 PENGTAOLI,QIXIANGYANG,ANDYUEPINGZHU Forv 1,let E = x:S˜ g(x)>2v . BywaveletcharacterizationofSobolevspaces, we v r ≥ have 2pvE Cn. LetE = Qvo,l,whereQv,l aredisjointmaximaldyadiccubeswith v v v N | | ≤ l Qv,l P∈1. LetF bethesetofdSyadiccubescontainedinQv,l butnotinE ,F = F v,l v+1 v v,l | |≤ l and F = Ω F . Let E = x Q,Q F and we can write also E = SQ0,l, 0 v 0 0 0 \ { ∈ ∈ } v 1 l where Q0,l areS≥disjoint maximal dyadic cubes in Ω. The related set F is defiSned as 0,l F = Q Q0,landQ F . 0,l 0 ⊂ ∈ Fornanyv 0,wewritego (x) = gε Φε (x)andg (x) = gε Φε (x). Then g (x)isadesi≥redcombinationv,latom.TQhj,kPi∈sFvc,lomj,kplejt,kestheproovf. QjP,k∈Fv j,k j,k (cid:3) v 3. Waveletcharacterizationofthemultiplierspaces Inthissection,weuseMeyerwaveletstocharacterizethemultiplierspacesXt (Rn).For r,p anyg Ht,p(Rn), let gΦ,ε = g(x), 2jn(Φε)2(2jx k) . Let Φ(x) be a functionsatisfying ∈ j,k − Φ(x) 0, Φ(x) C (B(0,1D)) and Φ(x)dx = 1. FEor any g Ht,p(Rn), define gΦ = ≥ ∈ 0∞ ∈ j,k g(x), 2jnΦ(2jx k) . ThefunctionsRpacesSt (Rn)andSΦ,t(Rn)aredefinedasfollows. − r,p r,p D E Definition3.1. Givenr >0,t 0andr+t<1< p<n/(r+t). ≥ (i)Wesay f(x) St (Rn)if f(x)= fε Φε (x)and ∈ r,p (ε,j,Pk)∈Λn j,k j,k p/2 2j(n+2t)gΦ,ε2 fε 2χ(2jx k) dx C, Z | j,k | | j,k| − ≤ whereg Ht+r,p(Rn)an(εd,Xj,kg)∈ΛHnr+t,p(Rn) 1. ∈ k k ≤ (ii)Wesay f(x) SΦ,t(Rn)if f(x)= fε Φε (x)and ∈ r,p (ε,j,Pk)∈Λn j,k j,k p/2 2j(n+2t)gΦ 2 fε 2χ(2jx k) dx C, Z | j,k| | j,k| − ≤ whereg Hr+t,p(Rn)an(dε,Xj,kg)∈ΛHnr+t,p(Rn) 1. ∈ k k ≤ NowwegiveawaveletcharacterizationofthemultiplierspaceXt (Rn). LetΦ0(x)and r,p Φε(x), ε E be the scaling function and wavelet functions, respectively. For (ε, j,k), n ∈ (ε, j,k ),(ε”, j,k ) Λ andl Zn,let ′ ′ ′ ′ ′ n ∈ ∈ aε,ε′ = Φ0 (x)Φε (x), Φε′ (x) j,k,l,j′,k′ j,k+l j,k j′,k′ D E and aε,ε,ε′′,0 = (Φε )2 2jnΦ(2jx k), Φε′′ (x) . j,k,0,j′,k′ j,k − − j′,k′ D E Furthermore,for0 s N,ε E ,l Znands+ ε ε + l ,0,let ′ n ′ ≤ ≤ ∈ ∈ | − | || aε,ε′,ε′′,s = Φε′(x)Φε (x), Φε′′ (x) . j,k,l,j′,k′ j,k j+s,2sk+l j′,k′ D E Thefollowinglemmaisobtainedin[13].
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