Analytic Partial Differential Equations

Preface
Contents
Part I Distributions and Analyticity in Euclidean
Space
Chapter 1 Functions and Differential Operators in
Euclidean Space
1.1 Basic Notation and Terminology
1.1.1 Multi-index notation, partial derivatives, Leibniz rules
1.2 Smooth, Real-analytic, Holomorphic Functions
1.2.1 Regular functions in open subsets of Euclidean space
1.2.2 Germs and the language of Sheaf Theory
1.3 Differential Operators with Smooth Coefficients
Chapter 2 Distributions in Euclidean Space
2.1 Basics on Distributions in Euclidean Space
2.1.1 Definitions. Support and singular support
2.1.2 Tempered distributions and their Fourier transforms
2.1.3 The C∞ wave-front set of a distribution
2.1.4 Action of differential operators on distributions
2.2 Sobolev Spaces
2.3 Distribution Kernels
2.4 Fundamental Solutions, Parametrix, Hypoelliptic PDOs
Chapter 3 Analytic Tools in Distribution Theory
3.1 Analytic Parametrices, Analytic Hypoellipticity
3.2 Ehrenpreis’ Cutoffs and Analytic Regularity of Distributions
3.2.1 Ehrenpreis cutoffs
3.2.2 Using Ehrenpreis cutoffs to prove local analyticity
3.3 Distribution Boundary Values of Holomorphic Functions
3.3.1 Function spaces in complex wedges
3.3.2 Derivatives and antiderivatives of holomorphic functions in wedges
3.3.3 Distribution boundary values of holomorphic functions in wedges
3.3.4 Representation of distributions as sums of boundary values
3.4 The FBI Transform of Distributions. An Introduction
3.4.1 The classical FBI transform and its inversion
3.4.2 The FBI transform and analyticity
3.5 The Analytic Wave-Front Set of a Distribution
3.5.1 The analytic wave-front set. Microlocal analytic hypoellipticity
3.5.2 Analytic wave-front sets and holomorphic extension
3.5.3 Analytic wave-front sets and Ehrenpreis’ cutoffs
Chapter 4 Analyticity of Solutions of Linear PDEs. Basic Results
4.1 Analyticity of Solutions of Elliptic Linear PDEs
4.1.1 Elliptic linear PDOs with constant coefficients
4.1.2 Elliptic linear PDOs with variable coefficients. Constructing a parametrix
4.1.3 Exploiting the Gårding inequality
4.1.4 A word about elliptic systems
4.2 Degenerate Elliptic Equations. Influence of Lower Order Terms
4.2.1 The associated elliptic system of PDEs and the special condition on the matrix B
4.2.2 Statement of the main theorem. Necessity of condition
4.2.3 Construction of a fundamental solution
4.2.4 The result for basal ?s
4.2.5 Completing the proof of Theorem 4.2.6
4.3 A Generalization of the Harmonic Oscillator
4.3.1 Exploiting a priori estimates
4.3.2 Differential operators akin to the harmonic oscillator operator that are not analytic hypoelliptic
4.A Appendix: Hermite’s Functions and the Schwartz Space
Chapter 5 The Cauchy–Kovalevskaya Theorem
5.1 A Nonlinear Ovsyannikov Theorem
5.1.1 Ovsyannikov analyticity in a scale of Banach spaces
5.1.2 The Ovsyannikov Theorem for a first-order analytic ODE
5.2 Application: the Nonlinear Cauchy–Kovalevskaya Theorem
5.2.1 Well-posed Cauchy problem for a holomorphic fully nonlinear PDE
5.2.2 Reduction to a quasilinear first-order square system
5.2.3 End of the proof of the Cauchy–Kovalevskaya Theorem
5.3 Applications to Linear PDE
5.3.1 Linear version of Theorem 5.2.4
5.3.2 The Ovsyannikov approach to linear PDEs
5.3.3 Distribution solutions of the abstract linear ODE
5.3.4 The Holmgren Theorem
5.3.5 An application: global solvability of elliptic linear PDEs with analytic coefficients
5.4 Application to Integrodifferential Cauchy Problems
5.4.1 Cauchy Problem for the Camassa–Holm Equation
Part II Hyperfunctions in Euclidean Space
Chapter 6 Analytic Functionals in Euclidean Space
6.1 Analytic Functionals in Complex Domains
6.2 Analytic Functionals in Cn
6.2.1 The Laplace–Borel transform
6.2.2 Convolution of analytic functionals
6.2.3 Analytic functionals carried by Rn
6.3 Analytic Functionals in Rn as Cohomology Classes
6.3.1 Analytic functionals in Rn defined by integrals over closed hypersurfaces
6.3.2 The one-dimensional case
6.3.3 The higher-dimensional cases
Chapter 7 Hyperfunctions in Euclidean Space
7.1 The Sheaf of Hyperfunctions in Euclidean Space
7.1.1 Definition based on analytic functionals
7.1.2 The sheaf of hyperfunctions in Rn
7.1.3 Division of hyperfunctions by analytic functions
7.1.4 Solvability of analytic linear PDEs in hyperfunctions
7.1.5 Singularity hyperfunctions in Euclidean space
7.2 Boundary values of holomorphic functions in wedges
7.3 The FBI Transform of Analytic Functionals
7.3.1 The FBI transform of an analytic functional. Definition and basic properties
7.3.2 Inversion of the FBI transform
7.4 Analytic Wave-front Set of a Hyperfunction
7.4.1 FBI transform and local analyticity
7.4.2 Hyperfunctions as sums of boundary values of holomorphic functions
7.4.3 Analytic wave-front set of a hyperfunction
7.4.4 Differential operators and the analytic wave-front set
7.5 Edge of the Wedge
7.5.1 The theorem of the Edge of the Wedge
7.5.2 One more characterization of the analytic wave-front set
7.5.3 The Holmgren Theorem for hyperfunctions
7.6 Microfunctions in Euclidean space
Chapter 8 Hyperdifferential Operators
8.1 Action on Holomorphic Functions and on Hyperfunctions
8.1.1 Hyperdifferential operators with constant coefficients
8.1.2 Hyperdifferential operators with variable coefficients. Definition and localness
8.1.3 Composition and transposition
8.1.4 Action of hyperdifferential operators on analytic functionals
8.1.5 Action of hyperdifferential operators on hyperfunctions
8.2 Local Representation of Hyperfunctions
8.2.1 A class of entire functions of infra-exponential type
8.2.2 Local representation of hyperfunctions in term of smooth functions
8.3 Elliptic Hyperdifferential Operators
8.3.1 Elliptic hyperdifferential operators. Definition
8.3.2 Analytic hypoellipticity in hyperfunctions of elliptic hyperdifferential operators with constant coefficients
8.4 Solvability of Constant Coefficients Hyperdifferential Equations
8.4.1 Two minimum principles for functions of infra-exponential type
8.4.2 Solvability in holomorphic functions of hyperdifferential equations with constant coefficients
8.4.3 Solvability in hyperfunctions of hyperdifferential equations with constant coefficients
Part III Geometric Background
Chapter 9 Elements of Differential Geometry
9.1 Regular Manifolds
9.1.1 Regular functions and maps in Euclidean space
9.1.2 Regular manifolds
9.1.3 Function algebras on regular manifolds
9.2 Fibre Bundles, Vector Bundles
9.2.1 Fibre bundles on a manifold
9.2.2 Vector bundles
9.2.3 Section spaces and differential operators
9.2.4 Associated sphere and projective bundles
9.3 Tangent and Cotangent Bundles of a Manifold
9.3.1 Tangent bundle
9.3.2 Cotangent and cosphere bundle
9.3.3 The tautological one-form
9.3.4 Differential of functions, of maps
9.3.5 Immersions and submersions
9.4 Differential Complexes and Grassman Algebras
9.4.1 Differential operators between vector bundles and their symbols. Differential complexes
9.4.2 Exterior algebra of a vector space, of a vector bundle
9.4.3 The Grassman algebra of a manifold
9.4.4 Exterior derivative and De Rham complex
9.4.5 The ∂ complex
Chapter 10 A Primer on Sheaf Cohomology
10.1 Basics on Sheaf Cohomology
10.1.1 Cohomology valued in a sheaf
10.1.2 Connecting homomorphism and long exact sequences
10.2 Fine Sheaves and Fine Resolutions
10.2.1 Fine sheaves on paracompact Hausdorff spaces
10.2.2 Fine resolutions and sheaf cohomology
10.2.3 Examples
10.3 Relative Sheaf Cohomology
10.3.1 Relative sheaf cohomology. Definition
10.3.2 Relative ?-cohomology classes carried by a real domain
10.3.3 Hyperfunctions in Euclidean space as relative cohomology classes
10.3.4 Microfunctions in Euclidean space as relative cohomology classes
10.4 Edge of the Wedge in (Co)homological Terms
Chapter 11 Distributions and Hyperfunctions on a Manifold
11.1 Distributions and Currents on a Manifold
11.1.1 Densities and distribution-densities
11.1.2 Currents on a manifold
11.1.3 The Bochner–Martinelli formulas
11.2 Plurisubharmonic functions and pseudoconvex domains
11.2.1 Subharmonic functions in an open subset ? of the plane
11.2.2 Plurisubharmonic functions in an open subset of Cn
11.2.3 Pseudoconvex domains. Global exactness of the ?
-complex
11.2.4 Plurisubharmonic quadratic forms in Cn
11.3 Hyperfunctions and Microfunctions in an Analytic Manifold
11.3.1 Hyperfunctions in an orientable C? manifold
11.3.2 Singularity hyperfunctions in a manifold
11.3.3 Microfunctions in a manifold
Chapter 12 Lie Algebras of Vector Fields
12.1 The Lie Algebra of Smooth Vector Fields
12.2 Integral Manifolds. Frobenius’ Theorem
12.2.1 Frobenius’ Theorem
12.2.2 A classical example
12.2.3 Application: Local integrability of C? complex involutive structures
12.3 Local Flow of a Regular Vector Field
12.4 Foliations Defined By Analytic Vector Fields
12.5 Systems of Vector Fields Generating Special Lie Algebras
12.5.1 Systems of finite type. Elliptic systems
12.5.2 Vector fields generating finite-dimensional Lie algebras
Chapter 13 Elements of Symplectic Geometry
13.1 Elements of Symplectic Algebra
13.1.1 Symplectic vector spaces
13.1.2 Lagrangian subspaces
13.1.3 The fundamental matrix
13.1.4 Linear groups
13.1.5 The symplectic group
13.2 The Metaplectic Group
13.2.1 Some “elementary” unitary transformations in Rn
13.2.2 Irreducible representations of the Heisenberg group
13.2.3 The metaplectic group
13.2.4 The differential of the metaplectic representation and Grushin’s operators
13.3 Symplectic Manifolds
13.3.1 Symplectic manifolds
13.3.2 Hamiltonian vector fields, Poisson bracket
13.3.3 Submanifolds of a symplectic manifold
13.3.4 The Darboux Theorem
13.3.5 The cotangent bundle as a symplectic manifold
13.3.6 Systems of functions of finite type
13.4 Involutive Systems of Functions of Principal Type
13.4.1 Involutive systems of functions of principal type
13.4.2 Involutive systems of eikonal equations of principal type
13.4.3 The Hamiltonian system and its first integrals
13.4.4 Quasilinear systems of first-order PDEs associated to the eikonal equations
13.4.5 Solving the eikonal equations
13.4.6 The homogeneous case
13.5 Real and Imaginary Symplectic Structures in Elements of Symplectic Geometry C2n
13.5.1 R and I terminology in complex Euclidean space
13.5.2 E-conjugation and the associated Hermitian form
13.5.3 Complex Lagrangian subspaces transversal to an R-Lagrangian subspace
13.5.4 Real quadratic forms and R-Lagrangian subspaces
13.5.5 A special Abelian subgroup of (?,C)
13.6 Real and Imaginary Symplectic Structures on Complex Manifolds
13.6.1 The R and I terminology in a complex-analytic manifold
13.6.2 Conormal bundles of nondegenerate smooth real hypersurfaces
13.6.3 Outer conormal bundles of boundaries of strictly pseudoconvex domains
Part IV Stratification of Analytic Varieties and Division of Distributions by Analytic Functions
Chapter 14 Analytic Stratifications
14.1 Analytic Stratifications and Stratifiable Sets
14.1.1 The regular and singular parts of a subset of an analytic manifold
14.1.2 Basics on local analytic stratifications
14.2 Analytic Subvarieties
14.2.1 Analytic Subvarieties. Definition. Basic properties
14.2.2 Regular and singular parts of analytic varieties
14.2.3 Basic properties of unions of submanifolds and subvarieties
14.2.4 The Whitney umbrella
14.3 The Weierstrass Theorems
14.3.1 TheWeierstrass theorems for functions
14.3.2 The ring C {z1, ..., zn}. The Weierstrass theorems for germs
14.3.3 Complex-analytic varieties
14.3.A Appendix: integral domains, unique factorization domains, Noetherian rings
14.4 Local Partitions of a Complex Hypersurface
14.5 Local Stratifications of a Real-Analytic Variety
14.5.1 Local partition of a real subvariety
14.5.2 Proof that the partition (14.5.13) is a stratification
14.5.3 Main theorem and direct consequences
14.5.4 The Whitney property
14.6 Semianalytic Sets
14.6.1 Semianalytic sets. Definition and basic properties
14.6.2 The regular and singular parts of a semianalytic set
Chapter 15 Division of Distributions by Analytic Functions
15.1 The Lojasiewicz Inequality
15.1.1 Functions with tempered growth or slow decay at the boundary
15.1.2 Preparatory lemmas
15.1.3 Proof of the Lojasiewicz inequality
15.2 Division of Distributions by Analytic Functions
15.2.1 Statement of the division theorem and strategy of the proof
15.2.2 Extendible distributions
15.2.3 The division theorem when dim V = 0
15.2.4 Division of distributions supported by smooth graphs
15.2.5 Regularly separated sets
15.2.6 Regular separation and extendibility of distributions
15.2.7 Regular separation in analytic stratifications
15.2.8 End of the proof of the division theorem
15.2.9 Application: Tempered solutions of PDEs with constant coefficients
15.3 Desingularization and Applications
15.3.1 The Hironaka Theorem. Statement for C? varieties
15.3.2 Application: a variant of the Lojasiewicz inequality
15.3.3 Application: reciprocals of analytic functions as distributions
15.3.4 Reciprocal of analytic functions. An alternate proof
15.3.5 Tempered fundamental solutions of linear PDEs with constant coefficients
15.A Appendix
15.A.1 Coverings by cubes
15.A.2 Partitions of unity subordinate to an open covering by cubes
Part V Analytic Pseudodifferential Operators and Fourier Integral Operators
Chapter 16 Elementary Pseudodifferential Calculus in the C∞ Class
16.1 Standard Pseudodifferential Operators
16.1.1 Amplitudes and the quantizing functor Op
16.1.2 Standard pseudodifferential operators. Smoothing operators
16.1.3 Action on Sobolev spaces
16.1.4 Effect of diffeomorphisms on pseudodifferential operators
16.1.5 Pseudodifferential operators on a manifold, between vector bundles
16.2 Symbolic Calculus
16.2.1 Standard symbols
16.2.2 Formal symbols
16.2.3 Composition of symbols
16.2.4 Elliptic symbols and their inverses
16.2.5 Pseudodifferential operators decrease the wave-front set
16.3 Classical symbols and classical pseudodifferential operators
16.3.1 Classical symbols
16.3.2 Classical pseudodifferential operators
16.3.3 Characteristic set of a classical pseudodifferential operator
16.3.4 The subprincipal symbol of a classical pseudodifferential operator
16.4 The Weyl Calculus in Euclidean Space
16.4.1 Tempered symbols and the OpW functor
16.4.2 Composition of Weyl operators
16.4.3 An example related to the Schrödinger equation
Chapter 17 Analytic Pseudodifferential Calculus
17.1 Analytic Pseudodifferential Operators
17.1.1 Asymptotically analytic multipliers
17.1.2 Pseudoanalytic amplitudes
17.1.3 Analytic pseudodifferential operators acting on distributions
17.1.4 Analyticity of the distribution kernel off the diagonal
17.1.5 Analytic regularizing operators
17.1.6 Composition of analytic pseudodifferential operators
17.1.7 The effect of analytic diffeomorphisms
17.1.8 Analytic pseudodifferential operators in an analytic manifold M
17.2 Symbolic Calculus
17.2.1 Pseudoanalytic symbols in Euclidean space, true and formal
17.2.2 From amplitudes to symbols
17.2.3 Analytic pseudodifferential operators are analytic pseudolocal
17.2.4 Symbolic calculus
17.2.5 Elliptic analytic pseudodifferential operators
17.2.6 Resolvent and functionals of analytic pseudodifferential operators of order zero
17.2.7 Classical analytic pseudodifferential operators
17.3 Analytic Microlocalization In Distribution Theory
17.3.1 Asymptotically analytic cutoff multipliers
17.3.2 Inverse Fourier transform of the cutoff gR
17.3.3 Using gR (D) to delimit the analytic wave-front set of a distribution
17.3.4 Analytic pseudodifferential operators decrease the analytic wave-front set of distributions
17.4 Action on Singularity Hyperfunctions
17.4.1 Action on analytic functionals
17.4.2 Action on singularity hyperfunctions
17.4.3 Analytic pseudodifferential operators decrease the analytic wave-front set of singularity hyperfunctions
17.5 Microdifferential Operators
Chapter 18 Fourier Integral Operators
18.1 Fourier Distribution Kernels in Euclidean Space
18.1.1 The prototype
18.1.2 Amplitudes in Euclidean space
18.1.3 Real phase-functions in Euclidean space
18.1.4 Making sense of oscillatory integrals
18.1.5 Action of pseudodifferential operators on Fourier distribution kernels
18.2 The Lagrangian Manifold Associated to a Phase-function
18.2.1 The Lagrangian submanifold associated to a phase-function. Definition
18.2.2 The wave-front set of a Fourier distribution kernel
18.2.3 Strongly nondegenerate phase-functions. Local symplectic graphs
18.3 Fourier Integral Operators. Basics
18.3.1 Fourier integral operators. Definition
18.3.2 The effect of FIOs on wave-front sets
18.3.3 Exponentiation of a skew-symmetric pseudodifferential operator
18.4 Reduction of the Fiber Variables
18.4.1 Reducing N. General phase-functions
18.4.2 Reducing N. Nondegenerate phase-functions
18.4.3 Equivalent phase-functions
18.4.4 FIOs that are pseudodifferential operators
18.5 Composition and Continuity of Fourier Integral Operators
18.5.1 Composition of FIOs
18.5.2 Composites with opposite phase-functions I
18.5.3 Continuity of FIOs. Elliptic FIOs
18.5.4 Composites with opposite phase-functions II. Amplitudes that have leading terms
18.5.5 Egorov’s Theorem
18.6 Globally Defined Fourier Integral Operators
18.6.1 From Lagrangian submanifolds to FIOs
18.6.2 Densities attached to a local symplectic graph
18.6.3 The principal symbol of an FIO
18.7 Principles of Analytic Fourier Integral Operators
18.A Appendix: Stationary Phase Formal Expansion
Part VI Complex Microlocal Analysis
Chapter 19 Classical Analytic Formalism
19.1 Formal Analytic Series
19.1.1 Formal analytic series and their realizations
19.1.2 Auxiliary classes. Globalization and germification
19.1.3 Classical analytic symbols
19.1.4 General formal analytic series
19.2 Classical Analytic Differential Operators of Infinite Order
19.2.1 Classical analytic differential operators of infinite order. Definition. Action on formal analytic series
19.2.2 Elliptic classical analytic differential operators of infinite order
19.2.3 Action on formal analytic functional series and on formal microfunction series
19.2.4 Conjugacy of classical analytic differential operators of infinite order
19.2.5 Correspondence between classical symbols and classical analytic differential operators of infinite order
19.3 The Complex Stationary Phase Formula
19.3.1 Complex Stationary Phase Formula. Statement
19.3.2 Complex Stationary Phase Formula. Proof
19.3.3 Complex stationary phase formula and classical analytic differential operators of infinite order
19.3.4 Action of L (z, θ, Dz , λ) on formal microfunction series
19.4 Symbolic Calculus and the KdV Hierarchy
19.4.1 The differential algebra of polynomials in an infinite sequence of variables
19.4.2 The variational derivative
19.4.3 Range of the variational derivative
19.4.4 Formal analytic series with coefficients in ?
19.4.5 Formal analytic series with coefficients in ?[h]
19.4.6 Evolution equations and conserved quantities
19.4.7 The stabilizer of the Sturm–Liouville symbol
19.4.8 The KdV hierarchy
19.4.9 First integrals of the KdV equations
Chapter 20 Germ Fourier Integral Operators in Complex Space
20.1 Analytic Symbols
20.1.1 Analytic symbols. Definitions
20.1.2 Analytic symbols of finite order and pseudoanalytic symbols
20.1.3 Analytic symbols and nonclassical differential operators of infinite order
20.2 Contours and Function Spaces
20.2.1 Contours of integration
20.2.2 Holomorphic function spaces defined by an exponent-weight
20.2.3 Representations of the identity operator in O(Φ) z◦
20.3 Sjöstrand Pairs
20.3.1 Sjöstrand pairs. Definition
20.3.2 Dual Sjöstrand pairs
20.3.3 The case of the canonical phase-function
20.4 Germ Fourier-like Transforms
20.4.1 Deforming the contours of integration
20.4.2 Germ Fourier-like transforms. Definition
20.4.3 Germ Fourier-like transforms of hyperfunctions
20.5 Sjöstrand Triads and Germ Fourier Integral Operators
20.5.1 Sjöstrand triads
20.5.2 Germ FIOs. Action on holomorphic functions
20.5.3 Deforming the contours of integration
20.5.4 Symplectic considerations
Chapter 21 Germ Pseudodifferential Operators in Complex
Space
21.1 Germ Pseudodifferential Operators
21.1.1 Germ pseudodifferential operators. Definition
21.1.2 Reducing the integrals
21.1.3 Germ Fourier transform of holomorphic functions and its inversion
21.1.4 Composition of germ Fourier-like transforms with opposite phase-functions
21.1.5 Composition of germ FIOs with opposite phase-functions
21.1.6 Pseudodifferential operators and differential operators of infinite order
21.2 Classical Germ Pseudodifferential Operators
21.2.1 Classical germ pseudodifferential operators
21.2.2 Classical germ pseudodifferential operators and differential operators of infinite order
21.2.3 Action on classical Fourier-like transforms
21.2.4 Action on classical germ FIOs
21.2.5 Inversion of classical elliptic germ Fourier-like transforms
21.2.6 Inversion of classical elliptic germ FIOs. Egorov’s theorem
21.3 Action on distributions
21.4 Action on Hyperfunctions and Microfunctions
21.4.1 Effective phase-functions
21.4.2 Action on holomorphic functions in wedges
21.4.3 Action on analytic functionals
21.4.4 Integrating AU μ
21.4.5 Action on hyperfunctions and mapping into singularity hyperfunctions
21.4.6 Effect on the analytic wave-front set. Action on singularity hyperfunctions and microfunctions
Chapter 22 Germ FBI Transforms
22.1 Germ FBI Transforms
22.1.1 FBI phase-functions. Definition
22.1.2 Sjöstrand pairs associated to an arbitrary FBI phase-function
22.1.3 Germ FBI-like transforms of holomorphic functions. Classical FBI-like transforms
22.2 Germ FBI Transforms of Distributions
22.2.1 FBI phase-functions at real points
22.2.2 Germ FBI-like transforms of integrable functions and of distributions
22.3 The Equivalence Theorem for Distributions
22.3.1 Difference of two FBI phase-functions
22.3.2 Transfer operators
22.3.3 Exploiting the transfer operators
22.3.4 Equivalence of classical elliptic germ FBI transforms of distributions with different phase-functions
Part VII Analytic Pseudodifferential Operators of
Principal Type
Chapter 23 Analytic PDEs of Principal Type. Local Solvability
23.1 Pseudodifferential Operators of Principal Type
23.1.1 Pseudodifferential operators of principal type. Definition
23.1.2 Microlocal normal forms
23.1.3 Null bicharacteristic curves, surfaces
23.1.4 Condition (P) for vector fields
23.1.5 Condition (P) for analytic pseudodifferential operators
23.2 Local Solvability of Analytic PDEs of Principal Type
23.2.1 Basic definition. A functional-analytic condition
23.2.2 Formal series solutions and approximate solutions of a linear PDE of principal type
23.2.3 Local solvability of analytic differential operators of principal type. Necessity of Condition (P)
23.2.4 Sufficiency of Condition (P). 1. A functional-analytic a priori estimate
23.2.5 Sufficiency of Condition (P). 2. Patching up the microlocal estimates
Chapter 24 Analytic PDEs of Principal Type. Regularity of the Solutions
24.1 A New Concept: Subellipticity
24.1.1 Subellipticity. Definition and basic properties
24.1.2 Subellipticity implies hypoellipticity
24.1.3 From microlocal to local subellipticity
24.2 Statement of the Main Theorem
24.3 Hypoellipticity Implies (Q)
24.4 Property (Q) Implies Subellipticity
24.4.1 Construction of a parametrix I. Complex phase-functions under a no-change of sign hypothesis
24.4.2 Application to the action of B (x,Dx′)
24.4.3 Construction of a parametrix II. Transport equations and formal series amplitudes
24.4.4 Microlocal parametrices under Hypotheses (A1) and (A2)
24.4.5 Consequences of Hypothesis (Q)
24.4.6 A class of operator-valued pseudodifferential operators
24.4.7 End of the proof that (Q) implies subellipticity
24.5 Analytic Hypoellipticity Implies (Q)
24.6 Property (Q) Implies Analytic Hypoellipticity
24.6.1 Construction of an analytic parametrix I. Complex phase-function and amplitudes
24.6.2 Formal and approximate solution of an operator equation
24.6.3 Estimate of the error resulting from the insertion of the cut-offs
24.6.4 Analyticity off the diagonal of the distribution kernel associated to the microlocal parametrix
24.6.5 End of the proof of the main theorem: (Q) implies (b)
24.7 The C∞ Situation
24.8 Propagation of Analytic Singularities
24.8.1 Preliminaries
24.8.2 Propagation of analytic singularities along a null bicharacteristic curve
24.8.3 Propagation of analytic singularities along a null bicharacteristic surface
24.8.4 Nonconfinement of the null bicharacteristic leaves and some consequences
24.8.5 Solvability in hyperfunctions of an analytic PDE of principal type
24.A Appendix: Properties of Real Polynomials in a Single Variable
24.B Appendix: Analytic Estimates of Exponential Amplitudes
Chapter 25 Solvability of Constant Vector Fields of Type (1,0)
25.1 C-Convexity and Global Solvability
25.1.1 C-convex domains in Cn
25.1.2 A pseudoconvex domain ? that is not C-convex
25.1.3 Global solvability of (1,0)-vector fields with constant coefficients
25.2 Local Solvability at the Boundary. First Steps
25.2.1 Local holomorphic solvability at the boundary. Basics
25.2.2 The positive definite case
25.2.3 Adapted defining functions
25.2.4 Further decomposition of the defining function
25.2.5 Necessary conditions for local holomorphic solvability at the boundary
25.2.6 A sufficient condition for local holomorphic solvability at the boundary
25.2.7 Z-pseudoconvexity and strict
25.2.8 Symplectic interpretation of Z-pseudoconvexity
25.3 Local Solvability at the Boundary. Final Characterization
25.3.1 A new condition: quasiconvexity
25.3.2 Local holomorphic solvability at the boundary when c1,1 = 1. Necessity of quasiconvexity
25.3.3 Local holomorphic solvability at an analytic boundary when c1,1 = 1. Sufficiency of Condition (QCX)
25.3.4 Summary and symplectic interpretation
25.3.5 Z-convexity at a boundary point when Z has constant coefficients
25.4 The Differential Complex. Generalities
25.4.1 The global differential complex defined by a system of constant vector fields of type (1,0)
25.4.2 The differential complex in germs at boundary points. Basics
25.A Appendix: Minima of Families of Plurisubharmonic Functions
25.A.1 Minima of real-valued continuous functions in real space
25.A.2 Minima of functions plurisubharmonic in complex space, quasiconvex in time
Chapter 26 Pseudodifferential Solvability and Property (Ψ)
26.1 Solvability: the Difference between Differential and Pseudodifferential
26.1.1 Mizohata pseudodifferential operators
26.2 Property (Ψ)
26.2.1 Property (Ψ), Definition
26.2.2 Invariance of (Ψ). Preparatory results
26.2.3 Invariance of (Ψ). Statement and proof
26.2.4 Microlocal solvability of P in distributions. Necessity of Condition (Ψ)
26.2.5 Microlocal solvability of P. in microfunctions. Trépreau’s Theorem
26.3 Microlocal Solvability in Distributions
26.3.1 The pseudodifferential operator under study
26.3.2 Phase-function attached to the pseudodifferential operator P
26.3.3 Properties of the exponent-weight πφ
26.3.4 Partition of U′ × ℭ entailed by Condition (Ψ)
26.3.5 A new formulation of Condition (Ψ)
26.3.6 Approximate microlocal parametrices
26.3.7 Convergence of A(ε)
26.3.8 Sufficiency of Condition (Ψ) for solvability of P in microdistributions
26.3.9 Open questions
Chapter 27 Pseudodifferential Complexes in Tube Structures
27.1 Pseudodifferential Complexes of Principal Type
27.1.1 Involutive Systems of Pseudodifferential Operators of Principal Type. Microlocal normal forms
27.1.2 The pseudodifferential complex defined by P
27.2 Tube Pseudodifferential Complexes
27.2.1 The tube system P and the associated (pseudo)differential complexes
27.2.2 Preliminary reductions of the tube system P
27.2.3 First integrals of a system of partially Hamiltonian vector fields
27.3 Phase-function and Amplitude
27.3.1 Fourier-like phase function
27.3.2 Amplitude
27.4 Approximate Homotopy Formulas
27.4.1 A sequence of Fourier integral operators with complex phase
27.4.2 The operators K(P) acting on differential forms
27.4.3 Basic bounds on K(p) ◦ f
27.4.4 Approximate homotopy formulas
27.5 Homotopy Formulas
27.5.1 A sufficient condition for convergence
27.5.2 Conditions Π (V, ξ) and (Ψ)
27.5.3 Consequences of Property Π (V, ℭ) for the analytic wave-front sets
27.6 Poincaré Lemmas
27.6.1 The effect of local regularity
27.6.2 Local Poincaré Lemma modulo analytic errors
27.6.3 Concluding remarks
References
Notation Index
Index