Quantization on Groups and Garding inequality

TL;DR

This paper introduces Wick's quantization on groups, explores its connection with Kohn-Nirenberg quantization, and provides a straightforward proof of Garding inequalities for symbolic pseudo-differential calculi on groups.

Contribution

It extends quantization methods to groups using representation theory and offers a new proof of Garding inequalities in this context.

Findings

Wick's quantization is defined on groups using the group Fourier transform.

A connection between Wick's and Kohn-Nirenberg quantizations is established.

A simple proof of Garding inequalities for symbolic pseudo-differential operators on groups is provided.

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.