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.