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Pseudo-differential operators with nonlinear quantizing functions
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 23 January 2019, pp. 103-130
- Print publication:
- February 2020
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In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form
where
$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$
$\tau :{\open R}^n\to {\open R}^n$ 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.
