Pseudo-differential operators with nonlinear quantizing functions

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.

26 pages

References in corpus (1)

• On Hadamard's global inverse function theorem (On global inversion of homogeneous maps)