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Pseudo-differential operators with nonlinear quantizing functions

Abstract

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.

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Copyright

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.

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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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References

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1 Bayer, D.. Bilinear time-frequency distributions and pseudodifferential operators. Thesis (Wien: Universität Wien, 2010).
2Calderón, A. and Vaillancourt, R.. On the boundedness of pseudo-differential operators. J. Math. Soc. Japan 23 (1971), 374378.
3Cappiello, M. and Toft, J.. Pseudo-differential operators in a Gelfand-Shilov setting. Math. Nachr. 290 (2017), 738755.
4Fischer, V. and Ruzhansky, M.. Quantization on nilpotent Lie groups. Progress in Mathematics, Vol. 314 (Birkhäuser, 2016). (open access book).
5Lerner, N.. Metrics on the phase space and non-selfadjoint pseudo-differetial operators (Birkhäuser, 2010).
6Mantoiu, M. and Ruzhansky, M.. Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups. Doc. Math. 22 (2017), 15391592.
7Ruzhansky, M. and Sugimoto, M.. On global inversion of homogeneous maps. Bull. Math. Sci. 5 (2015), 1318.
8Ruzhansky, M. and Sugimoto, M.. Global L 2-boundedness theorems for a class of Fourier integral operators. Comm. Partial Diff. Equ. 31 (2006), 547569.
9Ruzhansky, M. and Turunen, V.. Pseudo-differential operators and symmetries (Birkhäuser, 2010).
10Shubin, M. A.. Pseudodifferential operators and spectral theory (Springer, 2001).
11Stein, E. M.. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (Princeton: Princeton University Press, 1993).
12Taylor, M. E.. Partial differential equations II. Qualitative studies of linear equations, 2nd edn. Applied Mathematical Sciences, 116 (New York: Springer, 2011).
13Toft, J.. Matrix parameterized pseudo-differential calculi on modulation spaces. Generalized functions and Fourier analysis, 215–235, Oper. Theory Adv. Appl., 260, Adv. Partial Differ. Equ. (Basel) (Cham: Birkhäuser/Springer, 2017).

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Pseudo-differential operators with nonlinear quantizing functions

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