Quantization on Nilpotent Lie Groups.

• Superior document: Progress in Mathematics Series ; v.314
• Author: Fischer, Veronique.
• TeilnehmendeR: Ruzhansky, Michael.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2016.; ©2016.
• Year of Publication: 2016
• Edition: 1st ed.
• Language: English
• Series: Progress in Mathematics Series
• Physical Description: 1 online resource (568 pages)

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.