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1、Chapitre6Littlewood-PaleytheoryIntroductionThepurposeofthischapteristheintroductionbythistheorywhichisnothingbutaprecisewayofcountingderivativesusingthelocalizationinthefrequencyspace.Forinstance,ifwelooktothedispersiveestimateforthewaveequation,weseethatthis
2、hypothesisoflocalizationappearsnaturally.6.1LocalizationinfrequencyspaceTheverybasicideaofthistheoryconsistsinalocalizationprocedureinthefrequencyspace.Theinterestofthismethodisthatthederivatives(ormoregenerallytheFouriermul-tipliers)actinaveryspecialwayondis
3、tributionstheFouriertransformofwhichissupportedinaballoraring.Moreprecisely,wehavethefollowinglemma.6.1.1BernsteininequalitiesLemme6.1.1(oflocalization)LetCbearing,Baball.AconstantCexistssothat,foranynonnegativeintegerk,anysmoothhomogeneousfunctionσofdegreem,
4、anycoupleofreal(a,b)sothatb≥a≥1andanyfunctionuofLa,wehaveαk+1k+d(1−1)Suppu!⊂λB⇒sup$∂u$Lb≤Cλab$u$La;α=k−k−1kαk+1kSuppu!⊂λC⇒Cλ$u$La≤sup$∂u$La≤Cλ$u$La;α=kProofofLemma6.1.1Usingadilationofsizeλ,wecanassumeallalongtheproofthatλ=1.LetφbeafunctionofD(Rd)thevalueofwh
5、ichis1nearB.Asu!(ξ)=φ(ξ)u!(ξ),wecanwrite,ifgdenotestheinversefouriertransformofφ,∂αu=∂αg&u.ApplyingYounginequalitiestheresultfollowsthroughααα$∂g$Lc≤$∂g$L∞+$∂g$L12dα≤2$(1+
6、·
7、)∂g$L∞≤2$(Id−∆)d((·)αφ)$L1≤Ck+1.63Toprovethesecondassertion,letusconsiderafunctionφ"w
8、hichbelongstoD(Rd{0})thevalueofwhichisidentically1neartheringC.UsingthealgebraicUsingthefollowingalgebraicidentity#
9、ξ
10、2k=ξ2···ξ2j1jk1≤j1,···,jk≤d#=(iξ)α(−iξ)α,(6.1)
11、α
12、=kandstatinggdef=F−1(iξ)α
13、ξ
14、−2kφ"(ξ),wecanwrite,asαju!=φ"u!that#u!=(−iξ)αg!u,!α
15、α
16、=kwhichim
17、pliesthat#u=g&∂αu(6.2)α
18、α
19、=kandthentheresult.Thisprovesthewholelemma.6.1.2DyadicpartitionofunityNow,letusdefineadyadicpartitionofunity.Weshalluseitallalongthistext.Proposition6.1.1LetusdefinebyCtheringofcenter0,ofsmallradius3/4andgreatradius8/3.Itexiststworadia
20、lfunctionsχandϕthevaluesofwhichareintheinterval[0,1],belongingrespectivelytoD(B(0,4/3))andtoD(C)suchthat#∀ξ∈Rd,χ(ξ)+ϕ(2−jξ)=1,(6.3)j≥0#∀ξ∈Rd{0},ϕ(2−jξ)=1,(6.4)j∈Z
21、j−j%
22、≥2⇒Suppϕ(2−j·)∩Sup
