Preface
1 Introduction
Ⅰ Fourier Transformation and Pseudodifferential Operators
2 Fourier Transformation and Tempered Distributions
2.1 Definition and Basic Properties
2.2 Rapidly Decreasing Functions - 8(Rn)
2.3 Inverse Fourier Transformation and Plancherel's Theorem
2.4 Tempered Distributions and Fourier Transformation
2.5 Fourier Transformation and Convolution of Tempered Distributions.
2.6 Convolution on 8'(Rn) and Fundamental Solutions
2.7 Sobolev and Bessel Potential Spaces
2.8 Vector-Valued Fourier-Transformation
2.9 Final Remarks and Exercises
2.9.1 Further Reading
2.9.2 Exercises
3 Basic Calculus of Pseudodifferentiai Operators on Rn
3.1 Symbol Classes and Basic Properties
3.2 Composition of Pseudodifferential Operators: Motivation
3.3 Oscillatory Integrals
3.4 Double Symbols
3.5 Composition of Pseudodifferential Operators
3.6 Application: Elliptic Pseudodifferential Operators and Parametrices.
3.7 Boundedness on C∞/b(Rn) and Uniqueness of the Symbol
3.8 Adjoints of Pseudodifferential Operators and Operators in (x, y)-Form
3.9 Boundedness on L2(Rn) and L2-Bessel Potential Spaces
3.1 0 Outlook: Coordinate Transformations and PsDOs on Manifolds
3.1 1 Final Remarks and Exercises
3.1 1.1 Further Reading
3.1 1.2 Exercises
Ⅱ Singular Integral Operators
4 Translation Invariant Singular Integral Operators
4.1 Motivation
4.2 Main Result in the Translation Invariant Case
4.3 Calder6n-Zygmund Decomposition and the Maximal Operator
4.4 Proof of the Main Result in the Translation Invariant Case
4.5 Examples of Singular Integral Operators
4.6 Mikhlin Multiplier Theorem
4.7 Outlook: Hardy spaces and BMO
4.8 Final Remarks and Exercises
4.8.1 Further Reading
4.8.2 Exercises
5 Non-Translation Invariant Singular Integral Operators
5.1 Motivation
5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators
5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem
5.4 Kernel Representation of a Pseudodifferential Operator
5.5 Consequences of the Kernel Representation
5.6 Final Remarks and Exercises
5.6.1 Further Reading
5.6.2 Exercises
Ⅲ Applications to Function Space and Differential Equations
6 Introduction to Besov and Bessel Potential Spaces
6.1 Motivation
6.2 A Fourier-Analytic Characterization of Ho1der Continuity
6.3 Bessel Potential and Besov Spaces - Definitions and Basic Properties
6.4 Sobolev Embeddings
6.5 Equivalent Norms
6.6 Pseudodifferential Operators on Besov Spaces
6.7 Final Remarks and Exercises
6.7.1 Further Reading
6.7.2 Exercises
7 Applications to Elliptic and Parabolic Equations
7.1 Applications of the Mikhlin Multiplier Theorem
7.1.1 Resolvent of the Laplace Operator
7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols
7.1.3 Spectrum of a Constant Coefficient Differential Operator
7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem
7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces
7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces
7.3 Applications of Pseudodifferential Operators
7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators
7.3.2 Resolvents of Parameter-Elliptic Differential Operators
7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems
7.4 Final Remarks and Exercises
7.4.1 Further Reading
7.4.2 Exercises
Ⅳ Appendix
A Basic Results from Analysis
A.1 Notation and Functions on Rn
A.2 Lebesgue Integral and LP-Spaces
A.3 Linear Operators and Dual Spaces
A.4 Bochner Integral and Vector-Valued LP-Spaces
A.5 Frechet Spaces
A.6 Exercises
Bibliography
Index
