Weyl calculus on graded groups

Abstract

The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on ${\mathbb{R}}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following the work by Măntoiu and Ruzhansky (2017), using the Hörmander symbol classes $S_{\rho ,\delta }^m(G)$ introduced in the book by Fischer and Ruzhansky (2016). We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn–Nirenberg calculus, on ${\mathbb{R}}^n$ and on general graded groups (Fischer and Ruzhansky (2016)). Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the Gårding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups.
In the particular case of the Heisenberg group ${\mathbb{H}}_n$, we are able to answer the fundamental question of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on ${\mathbb{H}}_n$?
Among other things, we discuss and investigate an analog of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as a notion of noncommutative Poisson bracket for symbols in the setting of stratified groups.