Advances in Microlocal and Time-Frequency Analysis (Applied and Numerical Harmonic Analysis)

ANHA Series Preface
Preface
Contents
Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces
	1 Introduction
	2 Preliminaries
		2.1 Weight Functions
		2.2 Gelfand-Shilov Spaces
		2.3 Short Time Fourier Transforms and Gelfand-Shilov Spaces
		2.4 A Broad Family of Modulation Spaces
		2.5 Pseudo-Differential Operators
		2.6 Symbol Classes
	3 Continuity on Modulation Spaces for Pseudo-Differential Operators with Symbols of Infinite Order
	References
Hardy Spaces on Weighted Homogeneous Trees
	1 Introduction
	2 Weighted Homogeneous Trees
		2.1 Doubling and Local Doubling Properties
		2.2 Admissible Trapezoids and Calderón–Zygmund Sets
	3 The Maximal Function
	4 Hardy Spaces
		4.1 Equivalence of Spaces H1,p(μ) for p(1,∞]
		4.2 Real Interpolation Properties of H1(μ)
		4.3 Boundedness of Singular Integrals on H1(μ)
	References
The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces
	1 Introduction and Main Result
	2 Preliminaries
	3 Well-Posedness for a Scalar Schrödinger-Type Equation
	4 Well-Posedness for First Order Systems
	5 Proof of Theorem 1
	References
Cone-Adapted Shearlets and Radon Transforms
	1 Introduction
	2 Preliminaries
		2.1 Notation
		2.2 The Wavelet Transform
		2.3 The Shearlet Transform
		2.4 The Radon Transform
		2.5 The Radon Transform Intertwines Wavelets and Shearlets
	3 Cone-Adapted Shearlets and Radon Transforms
	4 Generalizations
	References
Linear Perturbations of the Wigner Transform and the Weyl Quantization
	1 Introduction
	2 Preliminaries
		2.1 Function Spaces
		2.2 Basic Properties of M1
		2.3 Bilinear Coordinate Transformations
		2.4 Partial Fourier Transforms
	3 Matrix-Wigner Distributions
		3.1 Connection to the Short-Time Fourier Transform
		3.2 Main Properties of the Transformation BA
		3.3 Cohen Class Members as Perturbations of the Wigner Transform
			3.3.1 Main Properties of the Cohen Class
	4 Pseudodifferential Operators
		4.1 Boundedness Results
			4.1.1 Operators on Lebesgue Spaces
			4.1.2 Operators on Modulation Spaces
		4.2 Symbols in the Sjöstrand Class
	References
About the Nuclearity of S(Mp) and Sω
	1 Introduction and Preliminaries
	2 Results for the Space S(Mp)
	3 Results for the Space Sω and Examples
	References
Spaces of Ultradifferentiable Functions of Multi-anisotropic Type
	1 Introduction
	2 The Space EM( Ω)
	3 The Space Es,Γ(Ω)
	4 The Space of Multi-Anisotropic Gevrey Vectors
	5 The Characterization of the Space Es,Γ(Ω)
	6 The Space EM,Γ( Ω)
	References
Comparison Principle for Non-cooperative Elliptic Systems and Applications
	1 Introduction
	2 Comparison Principle for Non-Cooperative Elliptic Systems
	3 Existence of Classical Solution for Linear Non-Cooperative Elliptic System
	References
On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation
	1 Introduction
	2 Definitions
	3 Reduction of a Certain Integral Equation
	4 Representation Theorem
	References
Decay Estimates and Gevrey Smoothing for a Strongly Damped Plate Equation
	1 Introduction
	2 Proof of Theorem 1
	References
Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation
	1 Introduction
	2 Notation
	3 Fundamental Solution and Decay Estimates
	Appendix
	References
On Density Operators with Gaussian Weyl Symbols
	1 Introduction
	2 Density Operators
		2.1 Basic Definitions
		2.2 Reduced Density Operators
		2.3 Gaussian Symbols
		2.4 A Lemma on Gaussians
		2.5 Partial Traces of Gaussian Density Operators
	3 Separability of Gaussian Density Operators
		3.1 The Notion of Separability
		3.2 The Gaussian Case: Necessary and Sufficient Conditions
	References
On the Solvability of a Class of Second Order Degenerate Operators
	1 Introduction
	2 The Mixed-Type Case
		2.1 Example
		2.2 Example
		2.3 Example
		2.4 Example: A Mildly Complex Case
	3 The Schrödinger-Type Case
		3.1 Example
		3.2 Example
		3.3 Example
	4 The Mixed-Schrödinger-Type Case
		4.1 Example
		4.2 Example
		4.3 Example
	5 Concluding Remarks
	References
Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms
	1 Introduction
	2 The Linear Estimates
		2.1 Examples
	3 Proof of Theorem 1
	References
Integrating Gauge Fields in the ζ-Formulation of Feynman\'s Path Integral
	1 Introduction
	2 The Free Real Scalar Quantum Field
		2.1 The ζ-Regularized Vacuum Expectation Values of Hn and Hζ
		2.2 The N→∞ Particle Limit
	3 Free Complex Scalar Quantum Fields
	4 The Dirac Field
	5 Coupling a Fermion of Mass m to Light in 1+1 Dimensions
	6 Conclusion
	References
A Class of Well-Posed Parabolic Final Value Problems
	1 Introduction
		1.1 Background: Phenomena of Instability
		1.2 Main Tool: Injectivity
		1.3 The Abstract Final Value Problem
	2 Proof of Theorem 1
	3 The Heat Problem with Final Time Data
		3.1 The Boundary Homogeneous Case
		3.2 The Inhomogeneous Case
	References
Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators
	1 Introduction
	2 The C*-Algebras SO and QC
		2.1 The C*-Algebra SO of Slowly Oscillating Functions
		2.2 The C*-Algebra QC of Quasicontinuous Functions
	3 The Banach Algebras Zp,w and Zp,wπ
	4 Muckenhoupt Weights
		4.1 Submultiplicative Functions and Their Indices
		4.2 Functions in VMO0(Gamma) and SO0(Gamma)
		4.3 A Subclass of Muckenhoupt Weights
		4.4 Weights Locally Equivalent to Slowly Oscillating Muckenhoupt Weights
	5 Mellin Pseudo-Differential Operators and Their Applications
		5.1 Boundedness and Compactness of Mellin Pseudo-Differential Operators
		5.2 Symbols of Mellin Pseudo-Differential Operators
		5.3 Applications of Mellin Pseudo-Differential Operators
	6 Localization of Muckenhoupt Weights Satisfying Condition (A)
	References
Carleman Regularization and Hyperfunctions
	1 Introduction
	2 Notations and Review of Hyperfunctions
	3 Carleman Regularization: Preparations
	4 The Regularization
	5 The Representation Theorems
	6 Representation of Real Analytic Functions
	References
Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space
	1 Introduction
	2 Statement of Results
	3 Examples and Remarks
	4 Definitions and Tools
		4.1 Pseudodifferential Operators with Limited Smoothness
			4.1.1 Mapping Properties
			4.1.2 Composition, Adjoint and Sharp Gårding\'s Inequality
	5 Proof
		5.1 Regularization
		5.2 Symbol Space
		5.3 Transformation to a First-Order System
		5.4 Diagonalization
			5.4.1 First Step of Diagonalization
			5.4.2 Second Step of Diagonalization
		5.5 Conjugation
		5.6 Conclusion
	6 Concluding Remarks
	References
Quantization and Coorbit Spaces for Nilpotent Groups
	1 Introduction
	2 Framework
	3 Weyl Systems, the Fourier-Wigner Transform
	4 Pseudo-Differential Operators
	5 Phase-Space Shifts
	6 Coorbit Spaces—A Short Overview
	References
On the Measurability of Stochastic Fourier Integral Operators
	1 Introduction
	2 Classical Theory of Oscillatory Integrals and FIOs
		2.1 Oscillatory Integrals
		2.2 Classical Theory of FIOs
	3 Stochastic Fourier Integral Operators
	4 Applications
	References
Convolution and Anti-Wick Quantisation on Ultradistribution Spaces
	1 Introduction
	2 Preliminaries
	3 Existence of S*-Convolution and D*-Convolution
	4 Convolution with the Gaussian Kernel
		4.1 The Fourier-Laplace Transform on Ultradistributions
		4.2 Convolution with the Gaussian Kernel
	5 Anti-Wick Quantisation
	6 An Extension of Convolution
	References
Exact Formulas to the Solutions of Several Generalizations of the Nonlinear Schrödinger Equation
	1 Introduction
	2 Proof of Theorem 1
	3 Proof of Theorem 2
	References
Dirichlet-to-Neumann Operator and Zaremba Problem
	1 Introduction
	2 Reduction of Neumann Conditions to the Boundary
	3 Reduction of Mixed Problems to the Boundary
	4 The Relationship to the Edge Calculus
	References
Extended Gevrey Regularity via the Short-Time Fourier Transform
	1 Introduction
		1.1 Basic Notation
	2 Preliminaries
		2.1 Extended Gevrey Regularity
		2.2 The Associated Function to the Sequence Mτ,σp=pτp σ
		2.3 Modulation Spaces
	3 Decay Properties of the STFT
	4 Wave Front Sets WFτ,σ and STFT
	References
Wiener Estimates on Modulation Spaces
	1 Introduction
	2 Preliminaries
		2.1 Gelfand-Shilov Spaces and Gevrey Classes
		2.2 Ordered, Dual and Phase Split Bases
		2.3 Invariant Quasi-Banach Spaces and Spaces of Mixed Quasi-Normed Spaces of Lebesgue Types
		2.4 Modulation and Wiener Spaces
		2.5 Classes of Periodic Elements
	3 Estimates on Wiener Spaces and Periodic Elements in Modulation Spaces
		3.1 Estimates of Wiener Spaces
		3.2 Wiener Estimates on Short-Time Fourier Transforms, and Modulation Spaces
		3.3 Periodic Elements in Modulation Spaces
	References
The Gabor Wave Front Set of Compactly Supported Distributions
	1 Introduction
	2 Preliminaries
	3 The Gabor and the Classical Wave Front Sets
	4 Propagation of Singularities for Schrödinger Equations
	References
Applied and Numerical Harmonic Analysis (99 volumes)