Pseudo-differential operators with nonlinear quantizing functions

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=int_{mathbb{R}^n}int_{mathbb{R}^n}e^{i(x-y)cdotxi}sigma(x+tau(y-x),xi)u(y)dydxi, $$ where $tau:mathbb{R}^ntomathbb{R}^n$ 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.