Pseudo-differential operators with nonlinear quantizing functions

TL;DR

This paper develops a calculus for pseudo-differential operators with nonlinear quantizing functions, generalizing classical quantizations and applying to analysis on nilpotent Lie groups, including examples from Heisenberg groups.

Contribution

It introduces a new calculus for pseudo-differential operators with nonlinear quantizations, extending classical frameworks to more general functions τ.

Findings

Unified framework for various quantizations including nonlinear cases

Application to analysis on nilpotent Lie groups

Examples from Heisenberg groups illustrating the theory

Abstract

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi, $$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and $\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $\tau$ and here we investigate the corresponding calculus in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.