Real Analysis Methods for Markov Processes: Singular Integrals and Feller Semigroups

Preface
Contents
1 Introduction and Main Results
	1.1 Formulation of the Problem
	1.2 Statement of Main Results
	1.3 Summary of the Contents
		1.3.1 Part I: A Short Course in Functional Analysis and Real Analysis
		1.3.2 Part II: Elements of Function Spaces
		1.3.3 Part III: Theory of Singular Integral Operators
		1.3.4 Part IV: Dirichlet Problems for Elliptic Differential Equations with Discontinuous Coefficients
		1.3.5 Part V: Oblique Derivative Problems for Elliptic Differential Equations with Discontinuous Coefficients
		1.3.6 Part VI: Construction of Feller Semigroups with Discontinuous Coefficients
		1.3.7 Part VII: Appendices
Part I A Short Course in Functional Analysis and Real Analysis
2 Elements of Functional Analysis
	2.1 Metric Spaces and the Contraction Mapping Principle
	2.2 Linear Operators and Functionals
	2.3 Quasinormed Linear Spaces
		2.3.1 Compact Sets
		2.3.2 Bounded Sets
		2.3.3 Continuity of Linear Operators
		2.3.4 Topologies of Linear Operators
		2.3.5 Product Spaces
	2.4 Normed Linear Spaces
		2.4.1 Linear Operators on Normed Spaces
		2.4.2 Method of Continuity
		2.4.3 Finite-Dimensional Spaces
		2.4.4 The Hahn–Banach Extension Theorem
		2.4.5 Dual Spaces
		2.4.6 Annihilators
		2.4.7 Dual Spaces of Normed Factor Spaces
		2.4.8 Bidual Spaces
		2.4.9 Weak Convergence
		2.4.10 Weak* Convergence
		2.4.11 Dual Operators
		2.4.12 Adjoint Operators
	2.5 Closed Operators
	2.6 Complemented Subspaces
	2.7 Compact Operators
	2.8 The Riesz–Schauder Theory
	2.9 Fredholm Operators
	2.10 Hilbert Spaces
		2.10.1 Orthogonality
		2.10.2 The Closest-Point Theorem and Applications
		2.10.3 Orthonormal Sets
		2.10.4 Adjoint Operators
	2.11 Continuous Functions on Metric Spaces
		2.11.1 The Ascoli–Arzelà Theorem
		2.11.2 The Stone–Weierstrass Theorem
3 Elements of Measure Theory and upper L Superscript pLp Spaces
	3.1 Measure Theory
		3.1.1 Measurable Spaces
		3.1.2 Measurable Functions
		3.1.3 Measures
		3.1.4 Signed Measures
		3.1.5 Borel and Radon Measures
		3.1.6 Lebesgue–Stieltjes Measures
		3.1.7 Lebesgue Measures
		3.1.8 Product Measures
		3.1.9 The nn-Dimensional Lebesgue Measure
		3.1.10 Integrals
		3.1.11 Fubini\'s Theorem
	3.2 upper L Superscript pLp Spaces
		3.2.1 Basic Properties of upper L Superscript pLp-Spaces
		3.2.2 End of Proof of the Completeness of upper L Superscript p Baseline left parenthesis upper X right parenthesisLp(X)
	3.3 Minkowski\'s Inequality for Integrals
	3.4 Hardy\'s Inequality
	3.5 The Generalized Hölder Inequality
	3.6 The Generalized Young Inequality
	3.7 Convolutions
		3.7.1 Approximations to the Identity
		3.7.2 Friedrichs\' Mollifiers
	3.8 Distribution Functions
	3.9 Marcinkiewicz\'s Interpolation Theorem
	3.10 Riesz Potentials
4 Elements of Real Analysis
	4.1 BMO Functions
	4.2 VMO Functions
	4.3 The Calderón–Zygmund Decomposition
	4.4 The Hardy–Littlewood Maximal Function
	4.5 The John–Nirenberg Inequality
	4.6 The Sharp Function and the Space BMO
	4.7 Spherical Harmonics
Part II Elements of Function Spaces
5 Harmonic Functions and Poisson Integrals
	5.1 Lipschitz Domains and Green\'s Formulas
	5.2 Harmonic Functions
	5.3 Poisson Kernels
	5.4 Poisson Integrals
	5.5 Manipulations of Harmonic Functions
6 Besov Spaces via Poisson Integrals
	6.1 Definition of Besov Spaces
	6.2 Various Norms of Besov Spaces
7 Sobolev and Besov Spaces
	7.1 Sobolev Spaces
		7.1.1 First Definition of Sobolev Spaces
		7.1.2 Second Definition of Sobolev Spaces
		7.1.3 Definition of General Sobolev Spaces
		7.1.4 Sobolev Imbedding Theorems
		7.1.5 The Rellich–Kondrachov Theorem
	7.2 Besov Spaces on the Boundary
	7.3 Trace Theorems
	7.4 VMO Functions Revisited
8 Maximum Principles in Sobolev Spaces
	8.1 Weak Maximum Principle
	8.2 Hopf\'s Boundary Point Lemma
	8.3 Strong Maximum Principle
Part III Theory of Singular Integral Operators
9 Elements of Singular Integrals
	9.1 Singular Integrals of Calderón and Zygmund
	9.2 The Case of Bounded Kernels
	9.3 The Case of Continuous Kernels
	9.4 The Hilbert Transform
	9.5 Equimeasurable Functions
	9.6 The Hilbert Transform (Continued)
	9.7 The Case of Odd Kernels
	9.8 Riesz Kernels
	9.9 The Case of Even Kernels
	9.10 The General Case
10 Calderón–Zygmund Kernels and Their Commutators
	10.1 Calderón–Zygmund Kernels
	10.2 Commutators of Calderón–Zygmund Kernels
	10.3 Proof of Estimate 10.15
11 Calderón–Zygmund Variable Kernels and Their Commutators
	11.1 Operators Having Calderón–Zygmund Variable Kernels
	11.2 Local Version of Theorems 11.2 and 11.3
Part IV Dirichlet Problems for Elliptic Differential Equations with Discontinuous Coefficients
12 Dirichlet Problems in Sobolev Spaces
	12.1 Formulation of the Dirichlet Problem
	12.2 Statement of Main Results (Theorems 12.1 and 12.2)
13 Calderón–Zygmund Kernels and Interior Estimates
	13.1 Interior Representation Formula for Solutions
	13.2 Local Interior Estimates
	13.3 Proof of Theorem 12.1
14 Calderón–Zygmund Kernels  and Boundary Estimates
	14.1 Boundary Representation Formula for Solutions
	14.2 upper L Superscript pLp Boundedness of Boundary Singular Integral Operators
		14.2.1 Boundary Singular Integral Operators
		14.2.2 Proof of Theorem 14.2
	14.3 upper L Superscript pLp Boundedness of Boundary Commutators
		14.3.1 End of Proof of Theorem 14.6
		14.3.2 Proof of Theorem 14.5
		14.3.3 upper L Superscript pLp Boundedness of Integral Operators with Positive Kernel
	14.4 Local Boundary Estimates
	14.5 Proof of Theorem 12.2
15 Unique Solvability of the Homogeneous Dirichlet Problem
	15.1 VMO Functions and Friedrichs\' Mollifiers
	15.2 Proof of Theorem 15.1
Part V Oblique Derivative Problems for Elliptic Differential Equations with Discontinuous Coefficients
16 Regular Oblique Derivative Problems  in Sobolev Spaces
	16.1 Formulation of the Oblique Derivative Problem
	16.2 Statement of Main Results (Theorems 16.1 and 16.2)
	16.3 The Oblique Derivative Problem (16.4) and the Distance Function
17 Oblique Derivative Boundary Conditions
	17.1 Construction of Auxiliary Functions
	17.2 Proof of Estimate (17.16)
18 Boundary Representation Formula  for Solutions
19 Boundary Regularity of Solutions
20 Proof of Theorems 16.1 and 16.2
	20.1 Proof of Theorem 16.1
	20.2 Proof of Theorem 16.2
Part VI Construction of Feller Semigroups with Discontinuous Coefficients
21 Markov Processes and Feller Semigroups
	21.1 Markov Processes and Transition Functions
		21.1.1 Definition of a Markov Process
		21.1.2 Markov Transition Functions
		21.1.3 Feller Transition Functions
		21.1.4 Path Functions of Markov Processes
		21.1.5 Strong Markov Processes
	21.2 Transition Functions and Feller Semigroups
	21.3 Feller Semigroups and Their Infinitesimal Generators
22 Feller Semigroups with Dirichlet Condition
	22.1 Formulation of the Dirichlet Problem
	22.2 Proof of Theorem 22.1
	22.3 Proof of Theorem 22.2
	22.4 Proof of Theorem 1.5
		22.4.1 The Space upper C 0 left parenthesis ModifyingAbove normal upper Omega With quotation dash right parenthesisC0(overlineΩ)
		22.4.2 End of Proof of Theorem 1.5
	22.5 Proof of Remark 1.6
	22.6 Notes and Comments
23 Feller Semigroups with an Oblique Derivative Condition
	23.1 Formulation of the Oblique Derivative Problem
	23.2 Proof of Theorem 23.1
	23.3 Proof of Theorem 23.2
	23.4 Proof of Theorem 1.3
	23.5 Proof of Remark 1.4
	23.6 Notes and Comments
24 Feller Semigroups and Boundary Value Problems
	24.1 Green Operators and Harmonic Operators
	24.2 General Boundary Value Problems
	24.3 General Existence Theorem for Feller Semigroups
	24.4 Proof of Remark 24.1
	24.5 Notes and Comments
25 Feller Semigroups with a First Order Ventcel\' Boundary Condition
	25.1 Proof of Theorem 1.1
	25.2 Notes and Comments
26 Concluding Remarks
	26.1 Functional Analytic Approach to Markov Processes
	26.2 Feller Semigroups and Pseudo-Differential and Singular Integral Operators
	26.3 Stochastic Analysis Methods for Ventcel\' Boundary Value Problem
Appendix A Change-of-Variables Formulas in the nn-Dimensional Lebesgue Integral
A.1 Half-open Intervals and Figures
A.2 Translation Invariance Formula for the Lebesgue Integral
A.3 Change-of-Variables Formula Under Linear Transformations
A.3.1 Left Elementary Transformations in Linear Algebra
A.3.2 Proof of Theorem A.7
A.3.3 End of Proof of Theorem A.7
A.4 Change-of-Variables Formula Under Diffeomorphisms
A.4.1 Proof of Theorem A.14
A.5 Integration in Polar Coordinates
Appendix B A Short Course to the Potential Theoretic Approach
B.1 Hölder Continuity and Hölder Spaces
B.2 Interior Estimates for Harmonic Functions
B.3 Hölder Regularity for the Newtonian Potential
B.4 Hölder Estimates for the Second Derivatives
B.5 Hölder Estimates at the Boundary
B.6 The General Case
Appendix  References
Index