distributions, fourier transforms and microlocal analysis

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1、DISTRIBUTIONS,FOURIERTRANSFORMSANDMICROLOCALANALYSISYu.SafarovNotation•suppfdenotesthesupportofthefunctionf,thatis,theclosureoftheset{x:f(x)6=0};•amulti-indexαisasetofnnon-negativeintegers,α:={α1,α2,...,αn};•ifα,βaremulti-indicesthen

2、α

3、:=α1+α2+···+αn,α!:=α1!α2!..

4、.αn!andα+β:={α1+β1,α2+β2,...,αn+βn};•Rndenotesthen-dimensionalEuclideanspaceandx=(x,x,...,x),12ny=(y,y,...,y),ξ=(ξ,ξ,...,ξ)areelementsofRn;12n12n•ifx∈Rnandαisamulti-indexthenxα:=xα1xα2...xαn,∂:=∂,12nxk√∂xkD:=−i∂,∂α:=∂α1∂α2...∂αn,Dα:=(−i)

5、α

6、∂α,wherei=−1;xkxkxx1x2x

7、nxx•C∞(Rn)isthelinearspaceofallinfinitelydifferentiablefunctionsonRn;•C∞(Rn)isthelinearspaceofallinfinitelydifferentiablefunctionsonRn0withcompactsupports.References[DS]M.DimassiandJ.Sj¨ostrand,Spectralasymptoticsinthesemi-classicalLimit,LMSLectureNotesSeries,vol.268

8、,CambridgeUniversityPress,Cambridge,1999.[H]L.H¨ormander,Theanalysisoflinearpartialdifferentialoperators.I-IV,Springer-Verlag,NewYork,1984.[Sh]M.Shubin,Pseudodifferentialoperatorsandspectraltheory,“Nauka”,Moscow,1978;Englishtransl.,Springer-Verlag,1987.[T]M.Taylor,

9、Pseudodifferentialoperators,PrincetonUniv.Press,Princeton,NewJersey,1981.[SV]Yu.SafarovandD.Vassiliev,Theasymptoticdistributionofeigenvaluesofpartialdifferen-tialoperators,AmericanMathematicalSociety,Providence,RhodeIsland,1996.TypesetbyAMS-TEX12Yu.SAFAROV1.Rapidly

10、decreasingfunctions1.SchwartzspaceS(Rn).Definition1.1.Wesaythatf∈S(Rn)iffisinfinitelydifferentiableandkfk:=sup

11、xβ∂αf(x)

12、<∞(1.1)α,βxx∈Rnforallmulti-indicesα,β.Obviously,S(Rn)isalinearspace.Iff∈S(Rn)then,forallmulti-indicesαandallpositiveintegersk,wehave

13、∂αf(x)

14、6c(1+

15、

16、x

17、)−kxα,kwithsomeconstantsc.Thereforethefunctionsf∈S(Rn)aresaidtoberapidlyα,kdecreasing.ItiseasytoseethatC∞(Rn)⊂S(Rn).02Example1.2.Thefunctionf(x)=e−

18、x

19、belongstoS(Rn).Lemma1.3.Iff∈S(Rn)thenxβ∂αf(x)∈S(Rn)andxXβαkx∂xfkα0,β06constkfkα00,β00

20、α00

21、6

22、α+α0

23、,

24、β00

25、6

26、β+β0

27、f

28、orallmulti-indicesα,β,α0,β0.Proof.Obvious.WeshallneedthefollowingversionofTaylor’sformula.Lemma1.4.Letmbeanon-negativeinteger,f∈S(Rn)andy∈Rnbeafixedpoint.Iffand