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內容簡介,圖書目錄,
內容簡介
圖書目錄
3.Wave Front and Operations on Distributions
3.1 The Trace of a Distribution.Product of Distnritbiaul Eiuation
3.2.The Wave Front of the Solution of a Differential Eqution
3.3.Wave Fronts and Integral Operators
Chapter 2.Pseudodifferential Operators
1.Algebra ofPseudodifferential Operators
1.1.Singular Integral Operators
1.2.The Symbol
1.3.Boundedness of Pseudodifferential Operators
1.4.Composition of Pseudodifferential Operators
1.5.The Formally Adjoint Operator
1.6.Pseudolocality.Microlocality
1.7.Elliptic Operators
1.8.GardingS Inequality
1.9.Extension 0f the Class of Pseudodifferential Operators
2.Invariance of the Principal SymboJ Under Canonical Transformations
2.1.Invariance Under the Change ofVariables.
2.2 The Subprincipal Symbol
2.3.Canonical Transformations
2.4.An Inverse Theorem
3.Canonical Forms ofthe Symbol
3.1.Simple Characteristic Points
3.2.Double Characteristics
3.3.The Complex-alued Symbol
3.4.The Canonical Form of the Symbol in a Neighbourhood of the Boundary.
4.Various Classes of Pseudodifferential Operators
4.1.The Lm/pδClasses
4.2.The Lm/φ，φ Classes
4.3.The Weyl Operators
5.Complex Powers ofElliptic Operators
5.1.The Definition ofComplex Powers.
5.2.Thc Construction of the Symbol for the Operator Az
5.3.The Construction of the Kernel of the Operator Az
5.4.The ξ-Function ofan Elliptic Operator
5.5.The Asymptotics of the Spectral Function and Eigenvalues
5.6.Complex Powers of an Elliptic Operator with Boundary Conditions
6.Pseudodifferential Operators in IRn and Quantization
6.1.The Analogy Between the Microlocal Analysis and the Quantization
6.2.Pseudodifierential 0perators in Rn
Chapter 3.Fourier Integral Operators
1.The Parametrix of the Cauchy Problem for Hyperbolic Equations
1.1.The Cauchy Problem for the Wave Equation
1.2.The Cauchy Problem for the Hyperbolic Equation of an Arbitrary 0rder.
1.3.The Method of Stationary Phase
2.The Maslov Canonical Operator
2.1.The MaslOV Index
2.2.Pre.canonieal Operator
2.3.The Canonical Operator
2.4.Some Applications.
3.Fourier Integral Operators
3.1.The Oscillatory Integrals
3.2.The Local Definition of the Fourier Integral Operator
3.3. The Equivalence of Phase Functions
3.4. The Connection with the Lagrange Manifold
3.5. The Global Definition of the Fourier Distribution
3.6. The Global Fourier Integral Operators
4. The Calculus of Fourier Integral Operators
4.1. The Adjoint Operator
4.2. The Composition of Fourier Integral Operators
4.3. The Boundedness in L2
5. The Image of the Wave Front Under the Action of a Fourier Integral Operator
5.1. The Singularities of Fourier Integrals
5.2. The Wave Front of the Fourier Integral
5.3. The Action of the Fourier Integral Operator on Wave Fronts
6. Fourier Integral Operators with Complex Phase Functions
6.1. The Complex Phase
6.2. Almost Analytic Continuation
6.3. The Formula for Stationary Complex Phase
6.4. The Lagrange Manifold
6.5. The Equivalence of Phase Functions
6.6. The Principal Symbol
6.7. Fourier Integral Operators with Complex Phase Functions
6.8. Some Applications
Chapter 4. The Propagation of Singularities
1. The Regularity of the Solution at Non-characteristic Points
1.1. The Microlocal Smoothness
1.2. The Smoothness of Solution at a Non-characteristic Point
2. Theorems on Removable Singularities
2.1. Removable Singularities in the Right-Hand Sides of Equations
2.2. Removable Singularities in Boundary Conditions
3. The Propagation of Singularities for Solutions of Equations of Real Principal Type
3.1. The Definition and an Example
3.2. A Theorem of H6rmander
3.3. Local Solvability
3.4. Semiglobal Solvability
4. The Propagation of Singularities for Principal Type Equations with a Complex Symbol
4.1. An Example
4.2. The Fixed Singularity
4.3. A Special Case
4.4. The Propagation of Singularities in the Case of a Complex Symbol of the General Form
5. Multiple Characteristics
5.1. Non-involutive Double Characteristics
5.2. The Levi Condition
5.3. Operators Having Characteristics of Constant Multiplicity .
5.4. Operators with Involutive Multiple Characteristics
5.5. The Schrrdinger Operator
Chapter 5. Solvability of (Pseudo)Differential Equations
1. Examples
1.1. Lewys Example
1.2. Mizohatas Equation
1.3. Other Examples
2. Necessary Conditions for Local Solvability
2.1. Hrrmanders Theorem
2.2. The Zero of Finite Order
2.3. The Zero of Infinite Order
2.4. Multiple Characteristics
3. Sufficient Conditions for Local Solvability
3.1. Operators of Real Principal Type
3.2. Operators of Principal Type
3.3. Operators with Multiple Characteristics
Chapter 6. Smoothness of Solutions of Differential Equations
1. Hypoelliptic Operators
1.1. Definition and Examples
1.2. Hypoelliptic Differential Operators with Constant Coefficients
1.3. The Gevrey Classes
1.4. Partially HypoeUiptic Operators
1.5. HypoeUiptic Equations in Convolutions
1.6. Hypoelliptic Operators of Constant Strength
1.7. Hypoelliptic Differential Operators with Variable Coefficients
1.8. Pseudodifferential Hypoelliptic Operators
1.9. Degenerate Elliptic Operators
1.10. Partial Hypoellipticity of Degenerate Elliptic Operators
1.11. Double Characteristics
1.12. Hypoelliptic Operators on the Real Line
2. Subeiliptic Operators
2.1. Definition and Simplest Properties
2.2. Estimates for First-Order Differential Operators with Polynomial Coefficients
2.3. Algebraic Conditions
……
Chapter 7 Transformation of Boundary-Value Problems
Chapter 8 Hyperfuctions
References
