Singular integrals and Feller semigroups : Real Analysis Methods for Markov Processes

講義名：解析学特別研究開設組織：博士課程数理物質科学研究科・数学専攻開講時期：2007年度（平成19年度）講義内容: 実解析学、関数解析学及び偏微分方程式論にまたがる基本的事項を通して、現代解析学の基本的な手法とアイデアに習熟することを目的とする。特に、確率論のマルコフ過程の関数解析的な研究を題材に選んで、実解析学、関数解析学、偏微分方程式論及び確率論の統合的な考察を行う。本文資料は英訳版である。(2020-11) Title: Singular integrals and Feller semigroupsSubtitle: Real Analysis Methods for Markov ProcessesContentsPreface page xiv1 Introduction and Main Results 11.1 Formulation of the Problem 11.2 Statement of Main Results 51.3 Summary of the Contents 81.4 A Bird\u27s Eye View of the Contents 16Part I A Short Course on Functional Analysis and Real Analysis 192 Elements of Functional Analysis 212.1 Metric Spaces and the Contraction Mapping Principle 212.2 Linear Operators and Functionals 232.3 Quasinormed Linear Spaces 242.3.1 Compact Sets 262.3.2 Bounded Sets 272.3.3 Continuity of Linear Operators 282.3.4 Topologies of Linear Operators 292.3.5 Product Spaces 292.4 Normed Linear Spaces 302.4.1 Linear Operators on Normed Spaces 312.4.2 Method of Continuity 332.4.3 Finite Dimensional Spaces 352.4.4 The Hahn{Banach Extension Theorem 352.4.5 Dual Spaces 382.4.6 Annihilators 382.4.7 Dual Spaces of Normed Factor Spaces 392.4.8 Bidual Spaces 392.4.9 Weak Convergence 402.4.10 Weak* Convergence 422.4.11 Dual Operators 422.4.12 Adjoint Operators 432.5 Closed Operators 442.6 Complemented Subspaces 472.7 Compact Operators 472.8 The Riesz{Schauder Theory 482.9 Fredholm Operators 502.10 Hilbert Spaces 522.10.1 Orthogonality 542.10.2 The Closest-Point Theorem and Applications 552.10.3 Orthonormal Sets 572.10.4 Adjoint Operators 582.11 Continuous Functions on Metric Spaces 602.11.1 The Ascoli-Arzela Theorem 602.11.2 The Stone-Weierstrass Theorem 612.12 Notes and Comments 623 Measures, Integration and Lp Spaces 633.1 Measure Theory 633.1.1 Measurable Spaces and Functions 633.1.2 Measures 653.1.3 Lebesgue Measures 663.1.4 Signed Measures 663.1.5 Borel Measures 673.1.6 Product Measures 683.1.7 Direct Image of Measures 683.1.8 Integrals 683.1.9 Fubini\u27s Theorem 703.2 Lp Spaces 753.3 Minkowski\u27s Inequality for Integrals 763.4 Hardy\u27s Inequality 813.5 The Generalized Holder Inequality 833.6 The Generelized Young Inequality 863.7 Convolutions 893.7.1 Approximations to the Identity 923.7.2 Friedrichs\u27 Mollifiers 963.8 Distribution Functions 993.9 Marcinkiewicz\u27s Interpolation Theorem 1013.10 Riesz Potentials 1053.11 Notes and Comments 1094 Elements of Real Analysis 1104.1 BMO Functions 1104.2 VMO Functions 1154.3 The Calderon-Zygmund Decomposition 1194.4 The Hardy-Littlewood Maximal Function 1274.5 The John-Nirenberg Inequality 1314.6 The Sharp Function and the Space BMO 1384.7 Spherical Harmonics 1454.8 Notes and Comments 166Part II Elements of Function Spaces 1675 Harmonic Functions and Poisson Integrals 1695.1 Lipschitz Domains and Green\u27s Formulas 1695.2 Harmonic Functions 1735.3 Poisson Kernels 1785.4 Poisson Integrals 1815.5 Manipulations of Harmonic Functions 1875.6 Notes and Comments 2006 Besov Spaces via Poisson Integrals 2016.1 Definition of Besov Spaces 2016.2 Various Norms of Besov Spaces 2046.3 Notes and Comments 2257 Sobolev and Besov Spaces 2267.1 Sobolev Spaces 2267.1.1 First Definition of Sobolev Spaces 2277.1.2 Second Definition of Sobolev Spaces 2277.1.3 Definition of General Sobolev Spaces 2307.1.4 Sobolev Imbedding Theorems 2307.1.5 The Rellich-Kondrachov Theorem 2317.2 Besov Spaces on the Boundary 2317.3 Trace Theorems 2357.4 VMO Functions Revisited 2357.5 Notes and Comments 2388 Maximum Principles in Sobolev spaces 2398.1 Mapping properties of Levy operators 2418.1.1 Boundedness of Levy operators defined on Rn 2418.1.2 The case of a bounded domain 2448.2 Weak Maximum Principle 2508.3 Hopf\u27s Boundary Point Lemma 2618.4 Strong Maximum Principle 2668.5 Notes and Comments 267Part III Theory of Singular Integral Operators 2699 Elements of Singular Integrals 2719.1 Singular Integrals of Calderon and Zygmund 2739.2 The Case of Bounded Kernels 2769.3 The Case of Continuous Kernels 2839.4 The Hilbert Transform 2979.5 Equimeasurable Functions 3019.6 The Hilbert Transform (Continued) 3119.7 The Case of Odd Kernels 3169.8 Riesz Kernels 3229.9 The Case of Even Kernels 3289.10 The General Case 3679.11 Notes and Comments 36910 Calderon-Zygmund Kernels and their Commutators 37010.1 Calderon-Zygmund Kernels 37010.2 Commutators of Calderon-Zygmund Kernels 37210.3 Proof of Estimate (10.6) 37410.4 Notes and Comments 38511 Calderon-Zygmund Variable Kernels and their Commutators 38611.1 Commutators of Calderon-Zygmund Variable Kernels 38611.2 Local Version of Theorems 11.1 and 11.2 39211.3 Notes and Comments 397Part IV Dirichlet Problems for Elliptic Differential Equations with Discontinuous Coefficients 39912 Dirichlet Problems in Sobolev Spaces 40112.1 Formulation of the Dirichlet Problem 40112.2 Statement of Main Results (Theorems 12.1and 12.2) 40312.3 Notes and Comments 40413 Calderon-Zygmund Kernels and Interior Estimates 40613.1 Interior Representation Formula for Solutions 40613.2 Local Interior Estimates 41113.3 Proof of Theorem 12.1 41413.4 Notes and Comments 42114 Calderon-Zygmund Kernels and Boundary Estimates 42214.1 Boundary Representation Formula for Solutions 42314.2 Lp Boundedness of Boundary Singular Integral Operators 42814.2.1 Boundary Singular Integral Operators 42914.2.2 Proof of Theorem 14.2 43514.3 Lp Boundedness of Boundary Commutators 43714.3.1 End of Proof of Theorem 14.6 44114.3.2 Proof of Theorem 14.5 45114.3.3 Lp Boundedness of Integral Operators with Positive Kernels 45814.4 Local Boundary Estimates 45814.5 Proof of Theorem 12.2 46314.6 Notes and Comments 46615 Unique Solvability of the Dirichlet Problem 46715.1 VMO Functions and Friedrichs\u27 Mollifiers 46815.2 Proof of Theorem 15.1 47115.3 Notes and Comments 483Part V Oblique Derivative Problems for Elliptic Differential Equations with Discontinuous Coefficients 48516 Oblique Derivative Problems in Sobolev Spaces 48716.1 Formulation of the Oblique Derivative Problem 48716.2 Statement of Main Results (Theorems 16.1and 16.2) 48916.3 Notes and Comments 49217 Oblique Derivative Boundary Conditions 49317.1 Construction of Auxiliary Functions 49317.2 Proof of Estimate (17.16) 50117.3 Notes and Comments 50418 Boundary Representation Formula for Solutions 50518.1 Integral representation formulas for the oblique derivative problem 50518.2 Notes and Comments 51819 Boundary Regularity of Solutions 51919.1 Boundary regularity of the solutions of problem (16.4) 51919.2 Notes and Comments 53620 Proof of Theorems 16.1and 16.2 53720.1 Proof of Theorem 16.1 53820.2 Proof of Theorem 16.2 54220.3 Notes and Comments 555Part VI Construction of Feller Semigroups with Discontinuous Coefficients 55721 Markov Processes and Feller Semigroups 55921.1 Markov Processes and Transition Functions 55921.1.1 Definition of a Markov Process 56021.1.2 Transition Functions 56221.1.3 Feller Transition Functions 56321.1.4 Path Functions of Markov Processes 56421.1.5 Strong Markov Processes 56521.2 Transition Functions and Feller Semigroups 56721.3 Feller Semigroups and their Infinitesimal Generators 56921.4 Notes and Comments 57122 Feller Semigroups with Dirichlet Condition 57222.1 Formulation of the Dirichlet Problem 57222.2 Proof of Theorem 22.1 57422.3 Proof of Theorem 22.2 57722.4 Proof of Theorem 1.3 58022.4.1 The Space C0(Ω) 58022.4.2 End of Proof of Theorem 1.3 58122.5 Proof of Remark 1.3 58522.6 Notes and Comments 58623 Feller Semigroups with Oblique Derivative Condition 58723.1 Formulation of the Oblique Derivative Problem 58723.2 Proof of Theorem 23.1 58923.3 Proof of Theorem 23.2 59323.4 Proof of Theorem 1.2 59623.5 Proof of Remark 1.2 60023.6 Notes and Comments 60124 Feller Semigroups and Boundary Value Problems 60324.1 Green Operators and Harmonic Operators 60424.2 General Boundary Value Problems 61124.3 General Existence Theorem for Feller Semigroups 61924.4 Proof of Remark 24.1 62324.5 Notes and Comments 62425 Proof of Theorem 1.1 62525.1 End of Proof of Theorem 1.1 62625.2 Notes and Comments 63726 Concluding Remarks 638Part VII Appendix 643Appendix 1 A Short Course to the Potential Theoretic Approach 645Bibliography 681Index 68