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 $$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$ where $\tau :{\open R}^n\to {\open R}^n$ is a general function. In particular, for the linear choices $\tau (x)=0$, $\tau (x)=x$ and $\tau (x)={x}/{2}$ 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 τ and here we investigate the corresponding calculus in the model case of ${\open R}^n$. We also give examples of nonlinear τ appearing on the polarized and non-polarized Heisenberg groups.