弱可微函数（影印版）

　　The term "weakly differentiable functions" in the title refers to those inte grable functions defined on an open subset of Rn whose partial derivatives in the sense of distributions are either Lr functions or （signed） measures with finite total variation. The former class of functions comprises what is now known as Sobolev spaces， though its origin， traceable to the early 1900s， predates the contributions by Sobolev. Both classes of functions， Sobolev spaces and the space of functions of bounded variation （BV functions）， have undergone considerable development during the past 20 years. From this development a rather complete theory has emerged and thus has provided the main impetus for the writing of this book. Since these classes of functions play a significant role in many fields， such as approximation theory， calculus of variations， partial differential equations， and non-linear potential theory， it is hoped that this monograph will be of assistance to a wide range of graduate students and researchers in these and perhaps other related areas. Some of the material in Chapters 1-4 has been presented in a graduate course at Indiana University during the 1987-88 academic year， and I am indebted to the students and colleagues in attendance for their helpful comments and suggestions.

暂缺《弱可微函数（影印版）》作者简介

Preface
1Preliminaries
1.1Notation
Innerproductofvectors
Supportofafunction
Boundaryofaset
Distancefromapointtoaset
Characteristicfunctionofaset
Multi-indices
Partialderivativeoperators
Functionspaces--continuous,HSldercontinuous,
HSldercontinuousderivatives
1.2MeasuresonRn
Lebesguemeasurablesets
LebesguemeasurabilityofBorelsets
Suslinsets
1.3CoveringTheorems
Hausdorffmaximalprinciple
Generalcoveringtheorem
Vitalicoveringtheorem
Coveringlemma,withn-ballswhoseradiivaryin
Lipschitzianway
Besicovitchcoveringlemma
Besicovitchdifferentiationtheorem
1.4HausdorffMeasure
EquivalenceofHausdorffandLebesguemeasures
Hausdorffdimension
1.5LP-Spaces
Integrationofafunctionviaitsdistribution
function
Young'sinequality
Holder'sandJensen'sinequality
1.6Regularization
LP-spacesandregularization
1.7Distributions
Functionsandmeasures,asdistributions
Positivedistributions
Distributionsdeterminedbytheirlocalbehavior
Convolutionofdistributions
Differentiationofdistributions
1.8LorentzSpaces
Non-increasingrearrangementofafunction
Elementarypropertiesofrearrangedfunctions
Lorentzspaces
O'Neil'sinequality,forrearrangedfunctions
EquivalenceofLP-normand(p,p)-norm
Hardy'sinequality
InclusionrelationsofLorentzspaces
Exercises
HistoricalNotes
2SobolevSpacesandTheirBasicProperties
2.1WeakDerivatives
Sobolevspaces
Absolutecontinuityonlines
LP-normofdifferencequotients
TruncationofSobolevfunctions
CompositionofSobolevfunctions
2.2ChangeofVariablesforSobolevFunctions
Rademacher'stheorem
Bi-Lipschitzianchangeofvariables
2.3ApproximationofSobolevFunctionsbySmooth
Functions
Partitionofunity
SmoothfunctionsaredenseinWk'p
2.4SobolevInequalities
Sobolev'sinequality
2.5TheRellich-KondrachovCompactnessTheorem
Extensiondomains
2.6BesselPotentialsandCapacity
RieszandBesselkernels
Besselpotentials
Besselcapacity
BasicpropertiesofBesselcapacity
CapacitabilityofSuslinsets
Minimaxtheoremandalternateformulationof
Besselcapacity
MetricpropertiesofBesselcapacity
2.7TheBestConstantintheSobolevInequality
Co-areaformula
Sobolev'sinequalityandisoperimetricinequality
2.8AlternateProofsoftheFundamentalInequalities
Hardy-Littlewood-Wienermaximaltheorem
Sobolev'sinequalityforRieszpotentials
2.9LimitingCasesoftheSobolevInequality
Thecasekp=nbyinfiniteseries
Thebestconstantinthecasekp=n
AnL-boundinthelimitingcase
2.10LorentzSpaces,ASlightImprovement
Young'sinequalityinthecontextofLorentzspaces
Sobolev'sinequalityinLorentzspaces
Thelimitingcase
Exercises
HistoricalNotes
3PointwiseBehaviorofSobolevFunctions
3.1LimitsofIntegralAveragesofSobolevFunctions
Limitingvaluesofintegralaveragesexceptfor
capacitynullset
3.2DensitiesofMeasures
3.3LebesguePointsforSobolevFunctions
ExistenceofLebesguepointsexceptforcapacity
nullset
Approximatecontinuity
Finecontinuityeverywhereexceptforcapacitynullset
3.4Lr-DerivativesforSobolevFunctions
ExistenceofTaylorexpansionsLp
3.5PropertiesofLP-Derivatives
ThespacesTk,tk,Tk'p,tk'p
TheimplicationofafunctionbeinginTk,patall
pointsofaclosedset
3.6AnLp-VersionoftheWhitneyExtensionTheorem
ExistenceofaGafunctioncomparabletothe
distancefunctiontoaclosedset
TheWhitneyextensiontheoremforfunctionsin
Tk'pandtk'p
3.7AnObservationonDifferentiation
3.8Rademacher'sTheoremintheLP-Context
AfunctioninTk'peverywhereimpliesitisin
tk'palmosteverywhere
3.9TheImplicationsofPointwiseDifferentiability
ComparisonofLP-derivativesanddistributional
derivatives
IfuEtk'P(x)foreveryx,andifthe
Lp-derivativesareinLp,thenuEWk'p
3.10ALusin-TypeApproximationforSobolevFunctions
IntegralaveragesofSobolevfunctionsareuniformly
closetotheirlimitsonthecomplementofsets
ofsmallcapacity
ExistenceofsmoothfunctionsthatagreewithSobolev
functionsonthecomplementofsetsof
smallcapacity
3.11TheMainApproximation
Existenceofsmoothfunctionsthatagreewith
Sobolevfunctionsonthecomplementofsetsof
smallcapacityandarecloseinnorm
Exercises
HistoricalNotes
4PoincareInequalities--AUnifiedApproach
4.1InequalitiesinaGeneralSetting
AnabstractversionofthePoincar6inequality
4.2ApplicationstoSobolevSpaces
Aninterpolationinequality
4.3TheDualofWm,p()
Therepresentationof(W0m,p())*
4,4SomeMeasuresin(W0m,p())*
Poincareinequalitiesderivedfromtheabstract
versionbyidentifyingLebesgueandHausdorff
measurewithelementsin(Wm,p())*
ThetraceofSobolevfunctionsontheboundaryof
Lipschitzdomains
Poincar6inequalitiesinvolvingthetraceof
aSobolevfunction
4.5Poincar6Inequalities
Inequalitiesinvolvingthecapacityoftheseton
whichafunctionvanishes
4.6AnotherVersionofPoincare'sInequality
Aninequalityinvolvingdependenceontheseton
whichthefunctionvanishes,notmerelyonits
capacity
4.7MoreMeasuresin(Wm,p())*
Sobolev'sinequalityforRieszpotentialsinvolving
measuresotherthanLebesguemeasure
Characterizationofmeasuresin(Wm,p(Rn))*
4.8OtherInequalitiesInvolvingMeasuresin(Wk,p)*
InequalitiesinvolvingtherestrictionofHausdorff
measuretolowerdimensionalmanifolds
4.9TheCasep=1
InequalitiesinvolvingtheL1-normofthegradient
Exercises
HistoricalNotes
5FunctionsofBoundedVariation
5.1Definitions
DefinitionofBVfunctions
Thetotalvariationmeasure‖Du‖
5.2ElementaryPropertiesofBVFunctions
Lowersemicontinuityofthetotalvariationmeasure
Aconditionensuringcontinuityofthetotal
variationmeasure
5.3RegularizationofBVFunctions
RegularizationdocsnotincreasetileBVnorm
ApproximationofBVfunctionsbysmoothfunctions
CompactnessinL1oftheUlfitballinBV
5.4SetsofFinitePerimeter
Definitionofsetsoffiniteperimeter
Tileperimeterofdomainswithsmoothboundaries
Isoperimetricandrelativeisoperimetricinequalityfor
setsoffiniteperimeter
5.5TheGeneralizedExteriorNormal
ApreliminaryversionoftheGauss-Greentheorem
Densityresultsatpointsofthereducedboundary
5.6TangentialPropertiesoftheReducedBoundaryandthe
Measure-TheoreticNormal
Blow-upatapointoftilereducedboundary
Themeasure-theoreticnormal
Thereducedboundaryiscontainedinthe
measure-theoreticboundary
Alowerboundforthedensityof‖DXE‖
Hausdorffmeasurerestrictedtothereducedboundary
isboundedaboveby‖DXE‖
5.7RectifiabilityoftheReducedBoundary
Countably(n-1)-rectifiablesets
Countable(n-1)-rectifiabilityofthe
measure-theoreticboundary
5.8TheGauss-GreenTheorem
TheequivalenceoftherestrictionofHausdorff
measuretothemeasure-theoreticboundary
and‖DXE‖
TheGauss-Greentheoremforsetsoffiniteperimeter
5.9PointwiseBehaviorofBVFunctions
Upperandlowerapproximatelimits
TheBoxinginequality
Thesetofapproximatejumpdiscontinuities
5.10TheTraceofaBVFunction
TheboundedextensionofBVfunctions
TraceofaBVfunctiondefinedintermsofthe
upperandlowerapproximatelimitsofthe
extendedfunction
Theintegrabilityofthetraceoverthe
measure-theoreticboundary
5.11Sobolev-TypeInequalitiesforBVFunctions
Inequalitiesinvolvingelementsin(BV())*
5.12InequalitiesInvolvingCapacity
Characterizationofmeasurein(BV())*
PoincareinequalityforBVfunctions
5.13GeneralizationstotheCasep>1
5.14TraceDefinedinTermsofIntegralAverages
Exercises
HistoricalNotes
Bibliography
ListofSymbols
Index