Table of Contents: Quantization on Nilpotent Lie Groups.

Quantization on Nilpotent Lie Groups.

Bibliographic Details
Main Author:	Fischer, Veronique.
Other Authors:	Ruzhansky, Michael.
Format:	eBook
Language:	English
Published:	Cham : Springer International Publishing AG, 2016.
Edition:	1st ed.
Series:	Progress in Mathematics Series
Subjects:	Electronic books.
Online Access:	Click to View

Table of Contents:

Intro
Preface
Contents
Introduction
Nilpotent Lie groups by themselves and as local models
Hypoellipticity and Rockland operators
Pseudo-differential operators
Quantization on homogeneous Lie groups and the book structure
Notation and conventions
Chapter 1 Preliminaries on Lie groups
1.1 Lie groups, representations, and Fourier transform
Representations
Haar measure
Fourier analysis
1.2 Lie algebras and vector fields
1.3 Universal enveloping algebra and differential operators
1.4 Distributions and Schwartz kernel theorem
1.5 Convolutions
Convolution of distributions
1.6 Nilpotent Lie groups and algebras
1.7 Smooth vectors and infinitesimal representations
1.8 Plancherel theorem
1.8.1 Orbit method
1.8.2 Plancherel theorem and group von Neumann algebras
Our framework
The Plancherel formula
Group von Neumann algebra
The abstract Plancherel theorem
1.8.3 Fields of operators acting on smooth vectors
Chapter 2 Quantization on compact Lie groups
2.1 Fourier analysis on compact Lie groups
2.1.1 Characters and tensor products
2.1.2 Peter-Weyl theorem
2.1.3 Spaces of functions and distributions on G
Distributions
Gevrey spaces and ultradistributions
2.1.4 lp-spages on the unitary dual G
2.2 Pseudo-differential operators on compact Lie groups
2.2.1 Symbols and quantization
2.2.2 Difference operators and symbol classes
2.2.3 Symbolic calculus, ellipticity, hypoellipticity
2.2.4 Fourier multipliers and Lp-boundedness
2.2.5 Sharp Garding inequality
Chapter 3 Homogeneous Lie groups
3.1 Graded and homogeneous Lie groups
3.1.1 Definition and examples of graded Lie groups
3.1.2 Definition and examples of homogeneous Lie groups
3.1.3 Homogeneous structure
Homogeneity
3.1.4 Polynomials.
3.1.5 Invariant differential operators on homogeneous Lie groups
3.1.6 Homogeneous quasi-norms
3.1.7 Polar coordinates
3.1.8 Mean value theorem and Taylor expansion
Taylor expansion
3.1.9 Schwartz space and tempered distributions
3.1.10 Approximation of the identity
3.2 Operators on homogeneous Lie groups
3.2.1 Left-invariant operators on homogeneous Lie groups
3.2.2 Left-invariant homogeneous operators
3.2.3 Singular integral operators on homogeneous Lie groups
3.2.4 Principal value distribution
3.2.5 Operators of type ν = 0
3.2.6 Properties of kernels of type ν, Re ν E [0,Q)
3.2.7 Fundamental solutions of homogeneous differential operators
3.2.8 Liouville's theorem on homogeneous Lie groups
Chapter 4 Rockland operators and Sobolev spaces
4.1 Rockland operators
4.1.1 Definition of Rockland operators
4.1.2 Examples of Rockland operators
4.1.3 Hypoellipticity and functional calculus
4.2 Positive Rockland operators
4.2.1 First properties
4.2.2 The heat semi-group and the heat kernel
4.2.3 Proof of the heat kernel theorem and its corollaries
4.3 Fractional powers of positive Rockland operators
4.3.1 Positive Rockland operators on Lp
4.3.2 Fractional powers of operators Rp
4.3.3 Imaginary powers of Rp and I + Rp
4.3.4 Riesz and Bessel potentials
4.4 Sobolev spaces on graded Lie groups
4.4.1 (Inhomogeneous) Sobolev spaces
4.4.2 Interpolation between inhomogeneous Sobolev spaces
4.4.3 Homogeneous Sobolev spaces
4.4.4 Operators acting on Sobolev spaces
4.4.5 Independence in Rockland operators and integer orders
4.4.6 Sobolev embeddings
Local results
Global results
4.4.7 List of properties for the Sobolev spaces
Properties of L2s(G)
4.4.8 Right invariant Rockland operators and Sobolev spaces
4.5 Hulanicki's theorem
4.5.1 Statement.
4.5.2 Proof of Hulanicki's theorem
First step
Second step
Main technical lemma
Last step
4.5.3 Proof of Corollary 4.5.2
Chapter 5 Quantization on graded Lie groups
5.1 Symbols and quantization
5.1.1 Fourier transform on Sobolev spaces
5.1.2 The spaces Ka,b(G), LL(L2a(G), L2b(G)), and L∞a,b(G)
5.1.3 Symbols and associated kernels
5.1.4 Quantization formula
5.2 Symbol classes Smρ,δ and operator classes Ψmρ,δ
5.2.1 Difference operators
5.2.2 Symbol classes Smρ,δ
5.2.3 Operator classes Ψmρ,δ
5.2.4 First examples
5.2.5 First properties of symbol classes
5.3 Spectral multipliers in positive Rockland operators
5.3.1 Multipliers in one positive Rockland operator
5.3.2 Joint multipliers
5.4 Kernels of pseudo-differential operators
5.4.1 Estimates of the kernels
Estimates at infinity
5.4.2 Smoothing operators and symbols
5.4.3 Pseudo-differential operators as limits of smoothing operators
5.4.4 Operators in Ψ0 as singular integral operators
5.5 Symbolic calculus
5.5.1 Asymptotic sums of symbols
5.5.2 Composition of pseudo-differential operators
5.5.3 Adjoint of a pseudo-differential operator
5.5.4 Simplification of the definition of Smρ,δ
5.6 Amplitudes and amplitude operators
5.6.1 Definition and quantization
5.6.2 Amplitude classes
5.6.3 Properties of amplitude classes and kernels
5.6.4 Link between symbols and amplitudes
5.7 Calderón-Vaillancourt theorem
5.7.1 Analogue of the decomposition into unit cubes
5.7.2 Proof of the case S00,0
5.7.3 A bilinear estimate
5.7.4 Proof of the case S0ρ,ρ
Strategy of the proof
5.8 Parametrices, ellipticity and hypoellipticity
5.8.1 Ellipticity
5.8.2 Parametrix
5.8.3 Subelliptic estimates and hypoellipticity
Local hypoelliptic properties
Global hypoelliptic-type properties.
Chapter 6 Pseudo-differential operators on the Heisenberg group
6.1 Preliminaries
6.1.1 Descriptions of the Heisenberg group
6.1.2 Heisenberg Lie algebra and the stratified structure
6.2 Dual of the Heisenberg group
6.2.1 Schródinger representations πλ
6.2.2 Group Fourier transform on the Heisenberg group
The Euclidean Fourier transform
The (Euclidean) Weyl quantization
The operator FHn(κ)(π1)
6.2.3 Plancherel measure
6.3 Difference operators Δxj and Δyj
6.3.1 Difference operators Δxj and Δyj
6.3.2 Difference operator Δt
6.3.3 Formulae
6.4 Shubin classes
6.4.1 Weyl-Hörmander calculus
6.4.2 Shubin classes Σmρ(Rn) and the harmonic oscillator
6.4.3 Shubin Sobolev spaces
6.4.4 The λ-Shubin classes Σmρ,λ(Rn)
6.4.5 Commutator characterisation of λ-Shubin classes
6.5 Quantization and symbol classes Smρ,δ on the Heisenberg group
6.5.1 Quantization on the Heisenberg group
6.5.2 An equivalent family of seminorms on Smρ,δ = Smρ,δ(Hn)
6.5.3 Characterisation of Smρ,δ(Hn)
6.6 Parametrices
6.6.1 Condition for ellipticity
6.6.2 Condition for hypoellipticity
6.6.3 Subelliptic estimates and hypoellipticity
Appendix A Miscellaneous
A.1 General properties of hypoelliptic operators
A.2 Semi-groups of operators
A.3 Fractional powers of operators
A.4 Singular integrals (according to Coifman-Weiss)
Calderón-Zygmund kernels on Rn
A.5 Almost orthogonality
A.6 Interpolation of analytic families of operators
Appendix B Group C* and von Neumann
B.1 Direct integral of Hilbert spaces
B.1.1 Convention: Hilbert spaces are assumed separable
B.1.2 Measurable fields of vectors
B.1.3 Direct integral of tensor products of Hilbert spaces
Definition of tensor products
Tensor products of Hilbert spaces as Hilbert-Schmidt spaces.
Direct integral of tensor products of Hilbert spaces
B.1.4 Separability of a direct integral of Hilbert spaces
B.1.5 Measurable fields of operators
B.1.6 Integral of representations
B.2 C*- and von Neumann algebras
B.2.1 Generalities on algebras
Algebra
Commutant and bi-commutant
Involution and norms
B.2.2 C*-algebras
B.2.3 Group C*-algebras
Reduced group C*-algebra
Pontryagin duality
B.2.4 Von Neumann algebras
B.2.5 Group von Neumann algebra
B.2.6 Decomposition of group von Neumann algebras and abstract Plancherel theorem
Schródinger representations and Weyl quantization
Explicit symbolic calculus on the Heisenberg group
List of quantizations
Bibliography
Index.

Quantization on Nilpotent Lie Groups.

Similar Items