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Quantization on Groups and Garding inequality

Lino Benedetto, Clotilde Fermanian Kammerer, Véronique Fischer

Abstract

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn-Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we give a simple proof of Garding inequalities for three globally symbolic pseudo-differential calculi on groups.

Quantization on Groups and Garding inequality

Abstract

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn-Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we give a simple proof of Garding inequalities for three globally symbolic pseudo-differential calculi on groups.
Paper Structure (40 sections, 27 theorems, 159 equations)

This paper contains 40 sections, 27 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Quantizations on groups
  3. Fourier analysis
  4. The dual set
  5. Fixing a Haar measure
  6. The Fourier transform
  7. The Plancherel Theorem
  8. The Kohn-Nirenberg quantization
  9. The space $L^2(G\times \widehat{G})$
  10. The quantization ${\rm Op}^{\rm KN}$ on $L^2(G\times \widehat{G})$
  11. Extension of ${\rm Op}^{\rm KN}$ to $\mathcal{C}(G, \mathcal{F} L^1(G))$
  12. Extension of ${\rm Op}^{\rm KN}$ via the inversion formula
  13. The Wick quantization
  14. The transformation $\mathcal{B}$
  15. The quantization ${\rm Op}^{\rm Wick}$
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $G$ be a connected compact Lie group. Let $m\in \mathbb R$. Assume that the symbol $\sigma\in S^{m}(G)$ satisfies the elliptic condition $\sigma \geq c_0 ({\rm id} +\widehat{\mathcal{L}})^{\frac{m}{2}}$ for some constant $c_0>0$. Then there exist constants $c,C>0$ such that

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • ...and 43 more