Sobolev spaces on metric measure spaces

448 Pages·2015·2.089 MB·English

by  Heinonen J., Koskela P., Shanmugalingam N.,  Tyson J.T.

SobolevSpacesonMetricMeasureSpaces
Analysisonmetricspacesemergedinthe1990sasanindependentresearchfieldproviding
aunifiedtreatmentoffirst-orderanalysisindiverseandpotentiallynonsmoothsettings.
Basedonthefundamentalconceptoftheuppergradient,thenotionofaSobolevfunction
wasformulatedinthesettingofmetricmeasurespacessupportingaPoincaréinequality.
Thiscoherenttreatmentfromfirstprinciplesisanidealintroductiontothesubjectfor
graduatestudentsandausefulreferenceforexperts.Itpresentsthefoundationsofthe
theoryofsuchfirst-orderSobolevspaces,thenexploresthegeometricimplicationsofthe
criticalPoincaréinequalityandindicatesnumerousexamplesofspacessatisfyingthis
axiom.Adistinguishingfeatureofthebookisitsfocusonvector-valuedSobolevspaces.
Thefinalchaptersincludeproofsofseverallandmarktheorems,includingCheeger’s
stabilitytheoremforPoincaréinequalitiesunderGromov–Hausdorffconvergenceandthe
Keith–Zhongself-improvementtheoremforPoincaréinequalities.
juhaheinonen(1960–2007)wasProfessorofMathematicsattheUniversityof
Michigan.Hisprincipalareasofresearchinterestincludedquasiconformalmappings,
nonlinearpotentialtheory,andanalysisonmetricspaces.Hewastheauthorofover60
researcharticles,includingseveralpublishedposthumously,andtwotextbooks.Amember
oftheFinnishAcademyofScienceandLetters,HeinonenreceivedtheExcellencein
ResearchAwardfromtheUniversityofMichiganin1997andgaveaninvitedlectureat
theInternationalCongressofMathematiciansinBeijingin2002.
pekkakoskelaisProfessorofMathematicsattheUniversityofJyväskylä,Finland.He
worksinSobolevmappingsandintheassociatednonlinearanalysis,andhehasauthored
over140publications.HegaveinvitedlecturesattheEuropeanCongressofMathematicsin
Barcelonain2000andattheInternationalCongressofMathematiciansinHyderabadin
2010.KoskelaisamemberoftheFinnishAcademyofScienceandLetters.Hereceivedthe
VäisäläAwardin2001andtheMagnusEhrnroothFoundationprizein2012.
nageswarishanmugalingamisProfessorofMathematicsattheUniversityof
Cincinnati.Herresearchinterestsincludeanalysisinmetricmeasurespaces,potential
theory,functionsofboundedvariation,andquasiminimalsurfacesinmetricsetting.She
developedthefoundationalpartofthestructureofSobolevspacesinmetricsettinginher
Ph.D.thesisin1999,andshehasalsocontributedtothedevelopmentofpotentialtheoryin
metricsetting.HerresearchcontributionswererecognizedbytheCollegeofArtsand
SciencesattheUniversityofCincinnatiwithaMcMickenDean’sAwardin2008.
jeremyt.tysonisProfessorofMathematicsattheUniversityofIllinoisat
Urbana-Champaign,workinginanalysisinmetricspaces,geometricfunctiontheory,and
sub-Riemanniangeometry.Hehasauthoredover40researcharticlesandcoauthoredtwo
otherbooks.TysonhasreceivedawardsforteachingfromtheUniversityofIllinoisatboth
thedepartmentalandcollegelevel.HeisaFellowoftheAmericanMathematicalSociety.
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1 M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2 J.B.GarnettandD.E.MarshallHarmonicMeasure
3 P.CohnFreeIdealRingsandLocalizationinGeneralRings
4 E.BombieriandW.GublerHeightsinDiophantineGeometry
5 Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6 S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7 A.ShlapentokhHilbert’sTenthProblem
8 G.MichlerTheoryofFiniteSimpleGroupsI
9 A.BakerandG.WüstholzLogarithmicFormsandDiophantineGeometry
10 P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
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12 J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13 M.GrandisDirectedAlgebraicTopology
14 G.MichlerTheoryofFiniteSimpleGroupsII
15 R.SchertzComplexMultiplication
16 S.BlochLecturesonAlgebraicCycles(2ndEdition)
17 B.Conrad,O.GabberandG.PrasadPseudo-ReductiveGroups
18 T.DownarowiczEntropyinDynamicalSystems
19 C.SimpsonHomotopyTheoryofHigherCategories
20 E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI
21 E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII
22 J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23 J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
24 E.RiehlCategoricalHomotopyTheory
Sobolev Spaces on Metric Measure Spaces
An Approach Based on Upper Gradients
JUHA HEINONEN
Late,UniversityofMichigan
PEKKA KOSKELA
UniversityofJyväskylä,Finland
NAGESWARI SHANMUGALINGAM
UniversityofCincinnati
JEREMY T. TYSON
UniversityofIllinoisatUrbana-Champaign
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NageswariShanmugalingamandJeremyT.Tyson2015
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Sobolevspacesonmetricmeasurespaces:anapproachbasedonuppergradients/
JuhaHeinonen,PekkaKoskela,NageswariShanmugalingam,JeremyT.Tyson.
pages cm.–(Newmathematicalmonographs)
Includesbibliographicalreferencesandindexes.
ISBN978-1-107-09234-1(Hardback)
1. Metricspaces. 2. Sobolevspaces. I. Heinonen,Juha. II. Koskela,Pekka.
III. Shanmugalingam,Nageswari. IV. Tyson,JeremyT.,1972–
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Contents
Preface pagexi
1 Introduction 1
2 Reviewofbasicfunctionalanalysis 7
2.1 Normedandseminormedspaces 7
2.2 Linearoperatorsanddualspaces 13
2.3 Convergencetheorems 16
2.4 Reflexivespaces 24
2.5 NotestoChapter2 34
3 LebesguetheoryofBanachspace-valuedfunctions 36
3.1 MeasurabilityforBanachspace-valuedfunctions 36
3.2 IntegrablefunctionsandthespacesLp(X :V) 42
3.3 Metricmeasurespaces 49
3.4 Differentiation 73
3.5 Maximalfunctions 90
3.6 NotestoChapter3 96
4 Lipschitzfunctionsandembeddings 98
4.1 Lipschitzfunctions,extensions,andembeddings 98
4.2 Lowersemicontinuousfunctions 107
4.3 Hausdorffmeasures 110
4.4 Functionswithboundedvariation 111
4.5 NotestoChapter4 119
5 Pathintegralsandmodulus 121
5.1 Curvesinmetricspaces 121
5.2 Modulusofacurvefamily 127
5.3 Estimatesformodulus 134
5.4 NotestoChapter5 141
vii
viii Contents
6 Uppergradients 143
6.1 Classicalfirst-orderSobolevspaces 143
6.2 Uppergradients 151
6.3 Mapswithp-integrableuppergradients 156
6.4 NotestoChapter6 166
7 Sobolevspaces 167
7.1 Vector-valuedSobolevfunctionsonmetricspaces 167
7.2 TheSobolevp-capacity 183
7.3 N1,p(X :V)isaBanachspace 190
7.4 ThespaceHN1,p(X :V)andquasicontinuity 198
7.5 MainequivalenceclassesandtheMECp property 201
7.6 NotestoChapter7 203
8 Poincaréinequalities 205
8.1 Poincaréinequalityandpointwiseinequalities 205
8.2 DensityofLipschitzfunctions 228
8.3 QuasiconvexityandthePoincaréinequality 233
8.4 Continuousuppergradientsandpointwise
Lipschitzconstants 238
8.5 NotestoChapter8 243
9 ConsequencesofPoincaréinequalities 245
9.1 Sobolev–Poincaréinequalities 245
9.2 LebesguepointsofSobolevfunctions 261
9.3 MeasurabilityofequivalenceclassesandMECp 271
9.4 Annularquasiconvexity 279
9.5 NotestoChapter9 282
10 OtherdefinitionsofSobolev-typespaces 285
10.1 TheCheeger–Sobolevspace 285
10.2 TheHajłasz–Sobolevspace 286
10.3 SobolevspacesdefinedviaPoincaréinequalities 290
10.4 TheKorevaar–Schoen–Sobolevspace 294
10.5 Summary 304
10.6 NotestoChapter10 304
11 Gromov–HausdorffconvergenceandPoincaréinequalities 306
11.1 TheGromov–Hausdorffdistance 306
11.2 Gromov’scompactnesstheorem 312
11.3 PointedGromov–Hausdorffconvergence 315
11.4 PointedmeasuredGromov–Hausdorffconvergence 324