Wavelets, Multiplier spaces and application to Schrödinger type operators with non-smooth potentials

0.27 MB·English

by  Pengtao Li

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)ifandonl(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.2εa∈nEdnX,Q2j.,k2⊂,Qthe mul|tipj,kl|ierspac−es Xt (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)/ZnQε1∈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(Tt)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)an(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)an(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].