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Pseudo-differential operators with nonlinear quantizing functions

Part of: Abstract harmonic analysis Miscellaneous topics - Partial differential equations Lie groups Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  23 January 2019

Massimiliano Esposito  and
Michael Ruzhansky
Massimiliano Esposito
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZUnited Kingdom ([email protected]; [email protected])
Michael Ruzhansky
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZUnited Kingdom ([email protected]; [email protected])
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Abstract

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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.

Keywords

Pseudo-differential operatorsquantizationsWeyl quantizationHeisenberg group

MSC classification

Primary: 35S05: Pseudodifferential operators
Secondary: 47G30: Pseudodifferential operators 43A70: Analysis on specific locally compact and other abelian groups 43A80: Analysis on other specific Lie groups 22E25: Nilpotent and solvable Lie groups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 103 - 130
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Royal Society of Edinburgh 2019

Footnotes

*

Current address: Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium and School of Mathematical Sciences, Queen Mary University of London, United Kingdom.

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