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### Pseudo-differential operators with nonlinear quantizing functions

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 Title: Pseudo-differential operators with nonlinear quantizing functions Authors: Esposito, MRuzhansky, M Item Type: Journal Article 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. Issue Date: 23-Jan-2019 Date of Acceptance: 1-Sep-2018 URI: http://hdl.handle.net/10044/1/62727 DOI: https://dx.doi.org/10.1017/prm.2018.148 ISSN: 0308-2105 Publisher: Cambridge University Press (CUP) Journal / Book Title: Proceedings of the Royal Society of Edinburgh: Section A Mathematics Copyright Statement: ©2019 The Royal Society of Edinburgh. This is an Open Access article, distributed under theterms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, providedthe original work is properly cited. Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)The Leverhulme Trust Funder's Grant Number: EP/R003025/1RPG-2017-151 Keywords: math.FAmath.FAmath.APmath.OA35S05, 47G30, 43A70, 43A80, 22E25math.FAmath.FAmath.APmath.OA35S05, 47G30, 43A70, 43A80, 22E250101 Pure Mathematics0102 Applied Mathematics Notes: 26 pages Publication Status: Published online Online Publication Date: 2019-01-23 Appears in Collections: Pure MathematicsMathematicsFaculty of Natural Sciences