Bruce K. Driver Contents
Analysis Tools with Applications
October 30, 2003 File:anal.tex Part I Back Ground Material
1 Introduction / User Guide ======3 1.1 Topologybeginnings...... 3 1.2 A Better Integral and an Introduction to Measure Theory .... 3
2 Set Operations ======7 2.1 Exercises...... 10
3 The Real and Complex Numbers ======11 3.1 TheRealNumbers ...... 12 3.1.1 TheDecimalRepresentationofaRealNumber...... 16 3.2 TheComplexNumbers...... 19 3.3 Exercises...... 20
4 Limits and Sums ======21 4.1 Limsups,LiminfsandExtendedLimits...... 21 4.2 Sumsofpositivefunctions...... 24 4.3 Sumsofcomplexfunctions...... 28 4.4 IteratedsumsandtheFubiniandTonelliTheorems...... 32 4.5 Exercises...... 34 4.5.1 LimitProblems...... 34 4.5.2 DominatedConvergenceTheoremProblems ...... 35
5 `p — spaces, Minkowski and Holder Inequalities ======39 5.1 Exercises ...... 44 Springer Part II Metric and Banach Space Basics Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo 4Contents Contents 5
6 Metric Spaces ======49 11 Compactness ======135 6.1 Continuity...... 51 11.1MetricSpaceCompactnessCriteria...... 136 6.2 CompletenessinMetricSpaces...... 53 11.2CompactOperators...... 142 6.3 Supplementary Remarks ...... 55 11.3 Local and —Compactness...... 144 6.3.1 WordofCaution...... 55 11.4FunctionSpaceCompactnessCriteria...... 146 6.3.2 RiemannianMetrics...... 56 11.5 Tychono ’sTheorem ...... 149 6.4 Exercises...... 57 11.6Exercises...... 152 11.6.1Ascoli-ArzelaTheoremProblems...... 153 7 Banach Spaces ======61 11.6.2 Tychono ’sTheoremProblem...... 154 7.1 Examples...... 61 7.2 BoundedLinearOperatorsBasics ...... 64 7.3 GeneralSumsinBanachSpaces...... 68 Part IV Topological Spaces II 7.4 Inverting Elements in O([) ...... 70 7.5 HahnBanachTheorem...... 71 12 Locally Compact Hausdor Spaces ======157 7.6 Exercises...... 75 12.1LocallycompactformofUrysohn’sMetrizationTheorem.....161 7.6.1 Hahn-BanachTheoremProblems...... 77 12.2PartitionsofUnity ...... 164 12.3 F0([) and the Alexanderov Compactification...... 169 8TheRiemannIntegral======79 12.4 More on Separation Axioms: Normal Spaces ...... 171 8.1 TheFundamentalTheoremofCalculus...... 83 12.5Stone-WeierstrassTheorem...... 174 8.2 Integral Operators as Examples of Bounded Operators ...... 85 12.6LocallyCompactVersionofStone-WeierstrassTheorem...... 178 8.3 Linear Ordinary Di erentialEquations...... 87 12.7Exercises...... 179 8.4 ClassicalWeierstrassApproximationTheorem...... 91 8.5 Iterated Integrals ...... 96 13 Baire Category Theorem ======183 8.6 Exercises...... 99 13.1 Metric Space Baire Category Theorem ...... 183 13.2 Locally Compact Hausdor Space Baire Category Theorem . . . 184 9 Hölder Spaces as Banach Spaces ======103 13.3Exercises...... 190 9.1 Exercises...... 107 14 Topological Vector Spaces ======191 14.1BasicFacts...... 191 Part III Topological Spaces: I 14.2 The Structure of Finite Dimensional Topological Vector Spaces 193 14.3MetrizableTopologicalVectorSpaces...... 195 10 Topological Space Basics ======113 10.1ConstructingTopologiesandCheckingContinuity...... 114 10.2ProductSpacesI ...... 120 Part V Calculus and Ordinary Di erential Equations in Banach 10.3Closureoperations...... 123 Spaces 10.4CountabilityAxioms...... 125 10.5Connectedness...... 127 15 Ordinary Di erential Equations in a Banach Space ======203 10.6Exercises...... 130 15.1Examples...... 203 10.6.1GeneralTopologicalSpaceProblems...... 130 15.2 Uniqueness Theorem and Continuous Dependence on Initial 10.6.2ConnectednessProblems...... 131 Data...... 205 10.6.3MetricSpacesasTopologicalSpaces...... 132 15.3LocalExistence(Non-LinearODE)...... 207 15.4GlobalProperties...... 210 15.5 Semi-Group Properties of time independent flows...... 216 15.6Exercises...... 218 6Contents Contents 7
16 Banach Space Calculus ======221 20 Fubini’s Theorem ======319 16.1 The Di erential...... 221 20.1 Fubini-Tonelli’s Theorem and Product Measure ...... 320 16.2ProductandChainRules...... 222 20.1.1 Application: More proofst.8.27 of the Weierstrass 16.3PartialDerivatives...... 225 Approximation Theorem 8.27...... 327 16.4SmoothDependenceofODE’sonInitialConditions ...... 226 20.2 Lebesgue measure on Rg ...... 330 16.5HigherOrderDerivatives...... 229 20.3PolarCoordinatesandSurfaceMeasure...... 333 16.6ContractionMappingPrinciple...... 233 20.4Exercises...... 337 16.7InverseandImplicitFunctionTheorems...... 235 16.8MoreontheInverseFunctionTheorem...... 239 21 Lp -spaces ======339 16.8.1 Alternate construction of j ...... 243 21.1Jensen’sInequality ...... 343 16.9Applications...... 243 21.2ModesofConvergence ...... 346 16.10Exercises...... 245 21.3 Completeness of Os —spaces...... 350 21.3.1Summary:...... 354 21.4ConverseofHölder’sInequality...... 355 Part VI Lebesbgue Integration Theory 21.5 Uniform Integrability ...... 361 21.6Exercises...... 367 17 Introduction: What are measures and why “measurable” sets ======251 22 Approximation Theorems and Convolutions ======369 17.1TheproblemwithLebesgue“measure”...... 252 22.1ConvolutionandYoung’sInequalities...... 374 22.1.1SmoothPartitionsofUnity...... 383 18 Measurability ======257 22.2Exercises...... 384 18.1 Algebras and —Algebras...... 257 18.2MeasurableFunctions...... 262 18.2.1Moregeneralpointwiselimits...... 269 Part VII Construction of Measures 18.3 —FunctionAlgebras ...... 269 18.4 Product —Algebras...... 276 23 Uniqueness and Regulrity of Measures ======389 18.4.1FactoringofMeasurableMaps ...... 279 23.1 Monotone Class and — Theorems...... 389 18.5Exercises...... 279 23.1.1Someotherproofsofpreviouslyprovedtheorems...... 392 23.2RegularityofMeasures...... 394 19 Measures and Integration ======281 19.1ExampleofMeasures ...... 284 24 Daniell — Stone Construction of Measures ======401 19.2 Integrals of Simple functions ...... 286 24.1 Finitely Additive Measures and Associated Integrals ...... 401 19.3 Integrals of positive functions ...... 287 24.1.1 Integrals associated to finitelyadditivemeasures ...... 403 19.4 Integrals of Complex Valued Functions ...... 295 24.2TheDaniell-StoneConstructionTheorem...... 406 19.5 Measurability on Complete Measure Spaces ...... 303 24.3ExtensionsofpremeasurestomeasuresI...... 410 19.6 Comparison of the Lebesgue and the Riemann Integral ...... 304 24.4RieszRepresentationTheorem...... 414 19.7DeterminingClassesforMeasures ...... 307 24.4.1 The Riemann — Stieljtes — Lebesgue Integral ...... 418 19.8 Appendix: Bochner Integral ...... 310 24.5Metricspaceregularityresultsresisted...... 421 19.9 Bochner Integrals (NEEDS WORK) ...... 314 24.6MeasureonProductsofMetricspaces...... 422 19.9.1 Bochner Integral Problems From Folland ...... 314 24.7 Measures on general infiniteproductspaces...... 424 19.10Exercises...... 316 24.8ExtensionsofpremeasurestomeasuresII...... 426 24.8.1 “Radon” measures on (R> ) Revisited...... 428 BR t.24.35 24.9 Supplement: Generalizations of Theorem 24.35 to Rq ...... 429 24.10Exercises...... 433 24.10.1The Laws of Large Number Exercises ...... 434 8Contents Contents 9
25 Daniell Integral Proofs ======437 30.5FredholmOperators ...... 554 25.1 Extension of Integrals ...... 438 30.6TensorProductSpaces ...... 560 25.2 The Structure of O1(L) ...... 446 25.3RelationshiptoMeasureTheory...... 447 31 Spectral Theorem for Self-Adjoint Operators ======565 31.1 —Algebras(overcomplexes)...... 572 26 Carathéodory’s Method of Constructing Measures ======453 31.1.1Exercises...... 576 26.1OuterMeasures...... 453 31.2TheSpectralTheorem...... 578 26.2 Carathéodory’s Construction Theorem ...... 455 31.2.1 Problems on the Spectral Theorem (Multiplication 26.3Regularityresultsrevisited ...... 459 OperatorForm)...... 585 31.3SpectralTheoryinHilbertSpace...... 588 27 Radon Measures and C0(X) ======463 31.3.1Multiplicationoperators...... 591 27.1MoreRegularityResults...... 466 31.3.2Spectrum...... 593 27.2TheRieszRepresentationTheorem...... 469t.53.11 31.3.3SpectralTheoremandtheFunctionalCalculus...... 595 27.2.1 Another proof of Theorem 27.11...... 473 31.3.4 Weak integrals of operator valued functionst.29.83...... 596 27.3 The dual of F0([) ...... 473 31.3.5 Fourier Transformt.29.83 Proof of Theorem 31.83...... 598 27.4 Special case of Riesz Theorem on [0> 1] ...... 477 31.3.6 Proof of Theorem 31.83...... 602 27.5Applications...... 478 31.3.7Extensionstocommutingself-adjointoperators ...... 603 27.6 The General Riesz Representation by Daniell Integrals ...... 481 31.3.8Exercises...... 611 27.7RegularityResults...... 483 31.4 Unbounded Self-Adjoint Operators ...... 612 27.8 Old Stu ...... 489 31.4.1 A!liatedoperators...... 622 27.8.1 Construction of measures on a simple product space. . . . 489 31.5BoundedSelf-AdjointOperators ...... 623 31.6TheStructureofAbelianBanachAlgebras...... 628
Part VIII Hilbert Spaces and Spectral Theory of Compact Operators Part IX Synthesis of Integral and Di erential Calculus
28 Hilbert Spaces ======495 32 Complex Measures, Radon-Nikodym Theorem and the 28.1HilbertSpacesBasics...... 495 Dual of Lp ======637 28.2HilbertSpaceBasis...... 504 32.1Radon-NikodymTheoremI...... 638 28.3FourierSeriesConsiderations ...... 507 32.2SignedMeasures ...... 644 28.4WeakConvergence...... 510 32.2.1HahnDecompositionTheorem...... 645 28.5 Supplement 1: Converse of the Parallelogram Law ...... 514 32.2.2JordanDecomposition...... 646 28.6 Supplement 2. Non-complete inner product spaces ...... 516 32.3ComplexMeasuresII ...... 650 28.7 Supplement 3: Conditional Expectation ...... 518 32.4AbsoluteContinuityonanAlgebra...... 654 28.8Exercises...... 521 32.5DualSpacesandtheComplexRieszTheorem...... 656 28.9FourierSeriesExercises ...... 524 32.5.1 Reflexivity...... 660 28.10 Dirichlet Problems on G ...... 528 32.6Exercises...... 661
29 Polar Decomposition of an Operator ======533 33 Lebesgue Di erentiation and the Fundamental Theorem of Calculus ======663 30 Compact Operators ======541 33.1 A Covering Lemma and Averaging Operators ...... 664 30.1HilbertSchmidtandTraceClassOperators...... 543 33.2MaximalFunctions...... 665 30.2 The Spectral Theorem for Self Adjoint Compact Operators . . . 548 33.3LebesqueSet...... 667 30.3StructureofCompactOperators...... 551 33.4TheFundamentalTheoremofCalculus...... 670 30.4TraceClassOperators...... 551 33.5 Alternative method to the Fundamental Theorem of Calculus . 681 10 Contents Contents 11 t.31.29 33.5.1 Proof of Theorem 33.29...... 683 38 Complex Di erentiable Functions ======789 33.6Examples:...... 683 38.1BasicFactsAboutComplexNumbers...... 789 33.7Exercises...... 684 38.2Thecomplexderivative...... 790 38.3 Contour integrals ...... 796 34 The Change of Variable Theorem ======t.32.1 687 38.4 Weak characterizations of K( ) ...... 803 34.1 Appendix: Other Approaches to proving Theorem 34.1 ...... 692 38.5SummaryofResults...... 807 34.2Sard’sTheorem...... 694 38.6Exercises...... 808 38.7ProblemsfromRudin...... 811 35 Surfaces, Surface Integrals and Integration by Parts ======699 35.1 Surface Integrals ...... 701 39 More Complex Variables: The Index ======813 35.2Moresphericalcoordinates...... 711 39.1UniqueLiftingTheorem...... 814 35.3 q — dimensional manifolds with boundaries ...... 715 39.2PathLiftingProperty...... 814 35.4DivergenceTheorem...... 718 39.2.1 Alternate Method ...... 816 35.5TheproofoftheDivergenceTheorem ...... 722t.33.23 35.5.1 The Proof of the Divergence Theorem 35.23 ...... 724 40 Residue Theorem ======825 35.5.2 Extensions of the Divergence Theorem to Lipschitz 40.1ResidueTheorem...... 825 domains...... 726 40.2OpenMappingTheorem ...... 828 35.6ApplicationtoHolomorphicfunctions...... 727 40.3ApplicationsofResidueTheorem...... 829 35.7 Dirichlet Problems on G ...... 730p.33.32 40.3.1Applications;fundamentaltheoryofAlgebra...... 830 35.7.1 Appendix: More Proofs of Proposition 35.32...... 734 40.4IsolatedSingularityTheory...... 831 35.8Exercises...... 736 41 Conformal Equivalence ======835
Part X Miracle Properties of Banach Spaces 42 Find All Conformal Homeomorphisms of V $ U ======839 42.1“SketchofProofofRiemannMapping”Theorem...... 840 36 Two Fundamental Principles of Banach Spaces ======741 42.1.1NormalFamilies...... 840 36.1TheOpenMappingTheorem ...... 741 36.1.1ApplicationstoFourierSeries...... 747 43 Littlewood Payley Theory ======849 36.2 Banach — Alaoglu’s Theorem ...... 750 43.0.2Applications...... 852 36.2.1WeakandStrongTopologies...... 750 36.2.2WeakConvergenceResults ...... 752 Part XII The Fourier Transform and Generalized Functions 36.3 Supplement: Quotient spaces, adjoints, and more reflexivity . . . 756 36.4Exercises...... 761 44 Fourier Transform ======859 36.4.1MoreExamplesofBanachSpaces ...... 761 44.1FourierTransform...... 860 36.4.2Hahn-BanachTheoremProblems...... 761 44.2SchwartzTestFunctions...... 863 36.4.3 Baire Category Result Problems ...... 761 44.3FourierInversionFormula ...... 865 36.4.4WeakTopologyandConvergenceProblems...... 762 44.4 Summary of Basic Properties of and 1 ...... 868 F F 37 Weak and Strong Derivatives ======763 44.5FourierTransformsofMeasuresandBochner’sTheorem...... 869 37.1 Basic DefinitionsandProperties...... 763 44.6 Supplement: Heisenberg Uncertainty Principle...... 873 37.2TheconnectionofWeakandpointwisederivatives...... 777 44.6.1Exercises...... 875 37.3Exercises...... 783 44.6.2MoreProofsoftheFourierInversionTheorem ...... 876
Part XI Complex Variable Theory 12 Contents Contents 13
45 Constant Coe!cient partial di erential equations ======879 53 L2 — Sobolev spaces on Rn ======989 45.1Ellipticexamples...... 880 53.1SobolevSpaces...... 989 45.2PoissonSemi-Group ...... 882 53.2Examples...... 999 q 45.3 Heat Equation on R ...... 883 53.3 Summary of operations on K4 ...... 1001 45.4 Wave Equation on Rq ...... 887 53.4 Application to Di erentialEquations...... 1004 45.5EllipticRegularity...... 893 53.4.1Dirichletproblem ...... 1004 45.6Exercises...... 898 54 Pseudo-Di erential Operators on Euclidean space ======1007 v 2p 46 Elementary Generalized Functions / Distribution Theory ==899 54.0.2 On the decay of C { and C{ { = ...... 1007 q h i | | 46.1 Distributions on X r R ...... 899 54.1Symbolsandtheiroperators...... 1009 46.2Examplesofdistributionsandrelatedcomputations...... 900 54.1.1AremarkontheFourierInversionFormula...... 1009 46.3Otherclassesoftestfunctions...... 908 54.2Amoregeneralsymbolclass...... 1011 46.4Compactlysupporteddistributions ...... 914 54.2.1Heuristics...... 1012 46.5TemperedDistributionsandtheFourierTransform...... 916 54.2.2Theproofs ...... 1014 46.6WaveEquation...... 924 54.3 Schwartz Kernel Approach ...... 1021 4 46.7 Appendix: Topology on Ff (X) ...... 929 54.4 Pseudo Di erentialOperators...... 1027
47 Convolutions involving distributions ======933 55 Elliptic Pseudo Di erential Operators on Rd ======1041 47.1TensorProductofDistributions...... 933 56 Pseudo di erential operators on Compact Manifolds ======1047 47.2EllipticRegularity...... 943t.45.4 47.3 Appendix: Old Proof of Theorem 47.4...... 947 57 Sobolev Spaces on M ======1051 57.1 Alternate Definition of Kn for n-integer ...... 1056 Part XIII An Introduction to Di erentiable Manifolds 57.2ScaledSpaces...... 1058 57.3GeneralPropertiesof“Scaledspace"...... 1059 48 Inverse Function Theorem and Embedded Submanifolds ===955 48.1EmbeddedSubmanifolds ...... 955 48.2Exercises...... 956 Part XV PDE Examples 48.3ConstructionofEmbeddedSubmanifolds...... 957 58 Some Examples of PDE’s ======1065 49 The Flow of a Vector Fields on Manifolds======959 58.1SomeMoreGeometricExamples ...... 1070
50 Co-Area Formula in Riemannian Geometry t.57.3 ======963 50.0.1 Formal Proof of Theorem 50.3 ...... 969 Part XVI First Order Scalar Equations 50.1 Special case of the Co-area formula when [ = R ...... 972 50.2 Di erentialGeometricVersionofCo-AreaFormula ...... 974 59 First Order Quasi-Linear Scalar PDE ======1073 59.1LinearEvolutionEquations...... 1073 51 Application of the Co-Area Formulas ======977 59.1.1 A 1-dimensional wave equation with non-constant 51.1ExistenceofDensitiesforPushForwardsofMeasures...... 977 coe!cients ...... 1080 51.2SobolevInequalitiesandIsoperimetricInequalities...... 980 59.2GeneralLinearFirstOrderPDE ...... 1082 59.3Quasi-LinearEquations ...... 1088 59.4DistributionSolutionsforConservationLaws ...... 1094 2 Part XIV L — Sobolev spaces and Pseudo Di erential Operators 59.5Exercises...... 1099
52 L2 — Sobolev spaces on T n ======987 14 Contents Contents 15
60 Fully nonlinear first order PDE ======1105 67 Heat Equation ======t.68.1 1235 60.1AnIntroductiontoHamiltonJacobiEquations...... e.61.17 1110 67.1 Extensions of Theorem 67.1...... 1237 60.1.1 Solving the Hamilton Jacobi Equation (60.17) by 67.2RepresentationTheoremandRegularity...... 1241 characteristics ...... 1110 67.3WeakMaxPrinciples ...... 1243 60.1.2 The connection with the Euler Lagrange Equations ....1111 67.4Non-UniquenessofsolutionstotheHeatEquation...... 1249 60.2GeometricmeaningoftheLegendreTransform...... 1117 67.5 The Heat Equation on the Circle and R ...... 1251
61 Cauchy — Kovalevskaya Theorem ======1119 68 Abstract Wave Equation ======1253 61.1PDECauchyKovalevskayaTheorem...... t.62.7 1124 68.1 Corresponding firstorderO.D.E...... 1254 61.2 Proof of Theorem 61.7...... 1129 68.2DuHamel’sPrinciple...... 1256 61.3Examples...... 1130 69 Wave Equation on Rn ======1259 69.1 q =1Case ...... 1260 Part XVII Elliptic ODE 69.1.1 Factorization method for q =1...... 1262 69.2 Solution for q =3 ...... t.70.4 1263 62 A very short introduction to generalized functions ======1135 69.2.1 Alternate Proof of Theorem 69.4...... 1265 69.3DuHamel’sPrinciple...... 1266 63 Elliptic Ordinary Di erential Operators ======1139 69.4 Spherical Means ...... 1266 63.1SymmetricEllipticODE ...... 1140 69.5Energymethods...... 1269 63.2GeneralRegular2ndorderellipticODE...... 1143 69.6WaveEquationinHigherDimensions...... 1271 63.3ElementarySobolevInequalities...... 1153 69.6.1Solutionderivedfromtheheatkernel...... 1271 63.4AssociatedHeatandWaveEquations...... 1157 69.6.2SolutionderivedfromthePoissonkernel ...... 1272 63.5ExtensionstoOtherBoundaryConditions...... 1159 69.7 Explain Method of descent q =2...... 1276 63.5.1 Dirichlet Forms Associated to (O> G(O)) ...... 1161
Part XIX Sobolev Theory Part XVIII Constant Coe!cient Equations 70 Sobolev Spaces ======1279 64 Convolutions, Test Functions and Partitions of Unity ======1169 70.1 Mollifications...... t.71.10 1281 64.1ConvolutionandYoung’sInequalities...... 1169 70.1.1 Proof of Theorem 70.10...... 1285 64.2SmoothPartitionsofUnity...... 1180 70.2 Di erencequotients...... 1287 70.3SobolevSpacesonCompactManifolds...... 1289 65 Poisson and Laplace’s Equation ======1183 70.4TraceTheorems...... 1293 65.1HarmonicandSubharmonicFunctions ...... 1189 70.5ExtensionTheorems...... 1298 65.2Green’sFunctions...... 1199 70.6Exercises...... 1302 65.3ExplicitGreen’sFunctionsandPoissonKernels ...... 1203 65.4Green’sfunctionforBall ...... 1207 71 Sobolev Inequalities ======1303 65.5 Perron’s Method for solving the Dirichlet Problem ...... 1212 71.1 Morrey’s Inequality ...... 1303 65.6 Solving the Dirichlet Problem by Integral Equations ...... 1217 71.2Rademacher’sTheorem ...... 1309 71.3Gagliardo-Nirenberg-SobolevInequality...... 1310 66 Introduction to the Spectral Theorem ======1219 71.4SobolevEmbeddingTheoremsSummary ...... 1316 66.1 Du Hammel’s principle again...... 1226 71.5CompactnessTheorems...... 1318 71.6FourierTransformMethod...... 1322 71.7Othertheoremsalongtheselines...... 1323 71.8Exercises...... 1325 16 Contents Contents 17
81 T. Coulhon Lecture Notes ======1409 Part XX Variable Coe!cient Equations 81.1WeightedRiemannianManifolds ...... 1409 81.2GraphSetting...... 1411 72 2nd order di erential operators ======1329 81.3BasicInequalities...... 1412 72.1Outlineoffutureresults...... 1333 81.4AScaleofInequalities ...... 1415 81.5Semi-GroupTheory...... 1419 73 Dirichlet Forms ======1335 73.1Basics...... 1335 Part XXII Heat Kernels on Vector Bundles 74 Unbounded operators and quadratic forms ======1339 74.1 Unbounded operator basics ...... 1339 82 Heat Equation on Rn ======1425 74.2 Lax-Milgram Methods ...... 1341 74.3 Close, symmetric, semi-bounded quadratic forms and 83 An Abstract Version of E. Levi’s Argument ======1427 self-adjointoperators...... 1344 84 Statement of the Main Results 1431 74.4Constructionofpositiveself-adjointoperators...... 1349 ======84.1TheGeneralSetup:theHeatEq.foraVectorBundle...... 1431 74.5 Applications to partial di erentialequations...... 1350 84.2 The Jacobian (M —function)...... 1432 75 Weak Solutions for Elliptic Operators======1353 84.3TheApproximateHeatKernels ...... 1433 84.4TheHeatKernelanditsAsymptoticExpansion...... 1434 76 Elliptic Regularity ======1357 t.85.7 t.85.10 85 ProofofTheorems84.7 and 84.10 76.1InteriorRegularity...... 1357 t.85.7 ======1437 76.2BoundaryRegularityTheorem...... 1360 85.1 Proof of Theorem 84.7...... t.85.10 1437 85.2 Proof of Theorem 84.10...... 1439 77 L2 — operators associated to ======1375 E 86 Properties of 77.1 Compact perturbations of the identity and the Fredholm ======p.87.1 1441 Alternative...... 1376 86.0.1 Proof of Proposition 86.1...... 1442 77.2 Solvability of Ox = i andpropertiesofthesolution...... 1378 86.0.2 On the Operator Associated to the Kernel ...... 1443 77.3InteriorRegularityRevisited...... 1382 t.85.4 c.85.6 87 ProofofTheorem84.4 and Corollary 84.6 1447 77.4ClassicalDirichletProblem...... 1383 c.85.6 ======87.1 Proof of Corollary 84.6...... 1447 77.5SomeNon-CompactConsiderations...... 1384 t.85.4 87.2 Proof of Theorem 84.4...... 1448 77.5.1HeatEquation...... 1386 77.5.2WaveEquation...... 1386 88 Appendix: Gauss’ Lemma & Polar Coordinates ======1453 88.1TheLaplacianofRadialFunctions...... 1454 78 Spectral Considerations ======1387 78.1GrowthofEigenvaluesI...... 1388 89 The Dirac Equation a la Roe’s Book ======1457 89.1KernelConstruction...... 1460 89.2AsymptoticsbySobolevTheory...... 1463 Part XXI Heat Kernel Properties 90 Appendix: VanVleck Determinant Properties ======1465 79 Construction of Heat Kernels by Spectral Methods ======1395 l.85.3 90.1 Proof of Lemma 84.3...... r.85.2 1465 79.1 Positivity of Dirichlet Heat Kernel by Beurling Deny Methods. 1399 90.2 Another Proof of Remark 84.2: The Symmetry of M({> |)=.....1467 90.3NormalCoordinates ...... 1468 80 Nash Type Inequalities and Their Consequences ======1401 18 Contents Contents 19 s.68 91 Miscellaneous ======p.92.1 1473 99.2.1 mores.74 section 67 stu ...... 1542 91.1 Jazzed up version ofe.92.3 Proposition 91.1...... 1473 99.3 Section 73...... 1545 91.1.1 Proof of Eq. (91.3) ...... p.84.1 1475 99.4Boundaryvalueproblems...... s.77 1549 91.1.2 Old proof of Proposition 83.1...... t.85.7 1475 99.5 Section 76...... p.77.10 1551 91.1.3 Old Stu related to Theorem 84.7...... 1479 99.5.1 Summary of the Proofl.77.12 of Proposition 76.10...... 1552 99.5.2 Old Proofss.78 of Lemma 76.12...... 1553 92 Remarks on Covariant Derivatives on Vector Bundles =====1481 99.6 Old Section 77...... 1555 99.7Moredomaincomments...... 1555 93 Spin Bundle Stu ======1485 99.8OLDCompactness...... 1557 99.9FourierTransformMethod...... 1557 94 TheCasewhereM = n ======1487 R 99.10Old Lax-Milgram Theorem ...... 1560 94.1 Formula involving s ...... 1487 99.11OldLocalregularity...... 1561 94.2 Asymptotics of a perturbed Heat Eq. on Rq ...... 1488
Part XXV Gaussian Measures Part XXIII PDE Extras 100 Infinite Dimensional Gaussian Measures ======1567 95 Higher Order Elliptic Equations ======1495 100.1Finite Dimensional Examples and Results ...... 1569 96 Abstract Evolution Equations ======1499 100.2Density Theorems ...... 1572 N 96.1 Basic DefinitionsandExamples...... 1499 100.3Product measures on R ...... 1575 96.2GeneralTheoryofContractionSemigroups...... 1502 100.4Basic InfiniteDimensionalResults...... 1577 100.5Guassian Measure for c2 ...... 1585 97 Solutions to Exercises 100.6Classical Wiener Measure...... 1589 s.60 ======1513 97.1 Section 59Solutions ...... 1513 100.7Basic Properties Wiener Measure ...... 1594 97.1.1OldExercisesSolutions ...... 1514 100.8The Cameron-Martin Space and Theorem ...... 1596 s.33 100.9Cameron-Martin Theorem ...... 1599 97.2 Section 35Solutionss.66 ...... 1517 97.3 Section 65Solutions ...... 1519 100.10Exercises...... 1603 100.11Gross’AbstractWienerSpaces...... 1604
Part XXIV Old PDE Stu Part XXVI Old Unused Analysis Material 98 Old Section Stu s.60 ======1523 98.1 Section 59...... 1523 101 Old Stu s.61 ======c.4 1615 98.2 Section 60...... 1525 s.62 101.1Chapter 4 ...... t.4.22 1615 98.3 Section 61...... 1527 s.64 101.1.1Old proofs of Tonnelli’st.4.23 Theorem 4.22...... 1615 98.4 Old Section 63 Stu ...... 1528 s.66 101.1.2Old Proof of theorem 4.23...... p.4.17 1617 98.5 Old Section 65s.70 Stu ...... 1528 101.1.3Old Proof of Proposition 4.17...... 1618 98.6 Old Section 69Now to the q =3Case:formallycompute:.....1531 101.1.4Oldc.6 Fubini — sum...... 1620 98.7 Probably delete the following stu ...... 1535 101.2Chapter 6 ...... 1621 101.3Compactness on metric spaces ...... 1621 99 A Little Distribution Theory ======1537 q s.67 101.4Compact Sets in R ...... 1624 99.1 Old Section 66...... 1538 c.8 p.67.8 101.5Section 8c.11 ...... 1626 99.1.1 Old Proof of Propositiont.67.2166.8...... 1538 101.6Chapter 11...... 1627 99.1.2 Special case of Theorem 66.21...... 1540 t.8.27 s.68 101.6.1Oldc.10 Proof of Theorem 8.27 for g =1...... 1627 99.2 Old Section 67...... 1540 101.7Old 10 Stu ...... 1628 20 Contents Contents 21 c.12 s.71 101.8Chapter 12...... 1628 101.35Section 70...... 1699 101.8.1Olds.15 Urysohn’s metrization Theorem ...... 1628 101.36Applicationtoregularity...... s.32 1699 101.9Section 15...... s.18 1634 101.37Old Section 34...... s.75 1701 101.10Section 18: ...... 1636 101.38Old Section 74...... t.75.10 1702 101.10.1Miscellaneouss.20 measurability results ...... 1636 101.38.1An alternate proof of partt.75.23 of Theorem 74.10...... 1702 101.11Section 20...... n.20.2 1638 101.38.2Old Proof of Theoremt.75.2374.23...... 1703 101.11.1Alternate prooft.20.3 of comment after Notation 20.2...... 1639 101.38.3ProofofTheorems.A 74.23...... 1704 101.11.2Old Theorem 20.3...... 1639 101.39Old Section A...... s.54.6 1707 101.11.3Z’sRegularityofmeasuresonmetricspaces...... 1641 101.40Old Section 100.6 ...... 1709 101.11.4OldRegularityandDensityResults...... 1644 101.41Miscalaneous Old Stu ...... 1710 101.11.5Ageneralregularityresult...... 1644 101.41.1ProductswithaFiniteNumberofFactors...... 1710 101.11.6Regularityofmeasuresonmetricspaces...... t.23.15 1645 101.11.7Low tech. proof of Theorem 23.9...... 1649 s.21 Part XXVII Solutions to Selected Exercises 101.12Section 21...... s.22 1651 101.13Section 22...... 1652 s.13 s.26 102 Section 13 Solutions ======1739 101.14Section 28...... s.26 1653 s.15 101.14.1Proofs for Section 28viaorthonormalbases...... 1655 102.1Section 15Solutionss.16 ...... 1739 s.28 102.2Section 16Solutions ...... 1740 101.15Section 30...... t.28.18 1656 101.15.1Old Proof of Theorem 30.18...... 1656 s.18 s.24 103 Section 18 Solutions ======1743 101.16Section 24...... 1660 s.19 101.16.1Additive measures on R ...... 1665 104 Section 19 Solutions ======1745 101.16.2Moreexercises...... 1666 s.20 s.25 104.1Section 20Solutionss.21 ...... 1750 101.17Section 25...... s.30 1667 104.2Section 21Solutions ...... 1758 101.18Section 32 old Stu ...... 1668 s.22 104.3Section 22Solutionss.24 ...... 1768 101.19Signedmeasures...... 1669 104.4Section 24Solutions ...... 1770 101.19.1OldRadon-Nikodymproof...... 1671 s.26 104.5Section 28Solutionss.30 ...... 1774 101.20TheTotalVariationonanAlgebrabyB...... 1673 104.6Section 32Solutions ...... 1780 101.21TheTotalVariationanAlgebrabyZ...... 1675 s.31 s.31 104.7Section 33Solutionss.34 ...... 1789 101.22Section 33...... 1677 104.8Section 36Solutions ...... 1793 101.23OldAbsoluteContinuity ...... 1678 104.8.1Other Folland Chapter 5 problems ...... 1809 101.24Appendix: Absolute Continuity on an algebra by Z. (Delete?) . 1678 104.9Size of c2 —spaces...... 1812 101.25OtherHahnDecompositionProofs...... 1678 104.10Bochner Integral Problems form chapter 5 of firstedition. ....1813 101.26Old Dual to Os —spaces...... 1679 s.35 s.101.18 104.11Section 37Solutionss.42 ...... 1816 101.27Section 101.18s.31.4 ...... 1682 104.12Section 44Solutions ...... 1818 101.28Section 33.4 ...... 1683 s.43 104.13Section 45Solutionss.53 ...... 1823 101.29Section??...... s.34 1685 104.14‘Section 27Solutions...... 1824 101.30Section 36...... s.35 1689 104.15ProblemsfromFollandSec.7 ...... 1825 101.31Section 37...... p.35.12 1693 104.16FollandChapter2problems...... 1828 101.31.1OldProofofPropositionl.71.2337.12 in the a special case. . . . 1693 104.17FollandChapter4problems...... 1829 101.31.2OldProofofLemma70.23...... p.35.27 1695 101.31.3PartsofoldproofofTheorems.42 37.27 ...... 1695 101.32Section 44...... s.43 1697 Part XXVIII Appendices 101.33Old Section 45...... p.43.5 1698 101.33.1OldproofofProposition45.5...... 1698 101.34...... 1699 22 Contents
A Multinomial Theorems and Calculus Results ======5 A.1 MultinomialTheoremsandProductRules...... 5 A.2 Taylor’sTheorem...... 7
B Zorn’s Lemma and the Hausdor Maximal Principle ======11
CNets======15
References ======19
Index ======21