microlocal analysis.pdf

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1、IntroductiontoMicrolocalAnalysisRichardMelroseMassachusettsInstituteofTechnologyE-mailaddress:rbm@math.mit.edu0.5B;Revised:27-1-2003;Run:February20,20031991MathematicsSubjectClassication.AllovertheshopContentsPreface7Introduction9Chapter1.TempereddistributionsandtheFouriertransform111.1.Schwartzte

2、stfunctions111.2.Lineartransformations131.3.Tempereddistributions141.4.Twobigtheorems151.5.Examples161.6.Twolittlelemmas171.7.Fouriertransform191.8.Dierentialoperators221.9.Radialcompactication231.10.Problems24Chapter2.PseudodierentialoperatorsonEuclideanspace272.1.Symbols272.2.Pseudodierential

3、operators312.3.Composition332.4.Reduction342.5.Asymptoticsummation352.6.Residualterms372.7.ProofofCompositionTheorem382.8.Quantizationandsymbols392.9.Principalsymbol402.10.Ellipticity422.11.Ellipticregularity442.12.TheLaplacian442.13.L2boundedness452.14.Squareroot462.15.ProofofBoundedness472.16.Sob

4、olevboundedness482.17.Consequences502.18.Polyhomogeneity512.19.Linearinvariance542.20.Problems54Chapter3.Isotropicandscatteringcalculi573.1.Isotropicoperators5734CONTENTS3.2.Scatteringoperators603.3.Theresidualalgebraisotropicalgebra623.4.Theresidualisotropicring623.5.Exponentialandlogarithm653.6.F

5、redholmproperty653.7.Theharmonicoscillator663.8.L2boundednessandcompactness693.9.Sobolevspaces713.10.Theresidualgroup733.11.Representations733.12.Symplecticinvarianceoftheisotropicproduct743.13.Complexorder763.14.Tracesontheresidualalgebra773.15.Fredholmdeterminant793.16.Fredholmalternative833.17.R

6、esolventandspectrum843.18.Residuetrace853.19.Exteriorderivation883.20.Regularizedtrace893.21.Projections903.22.Complexpowers903.23.Indexandinvertibility903.24.Variation1-form923.25.Determinantbundle943.26.Indexbundle953.27.Indexformul953.28.Isotropicessentialsupport953.29.Isotropicwavefrontset953.

7、30.IsotropicFBItransform953.31.Problems95Chapter4.Microlocalization994.1.Calculusofsupports994.2.Singularsupports1004.3.Pseudolocality1004.4.Coordinateinvariance1014.5.Problems1034.6.Characteristicvariety10