Abstract
The purpose of this note is to compare the properties of the symbolic pseudo-differential calculus on the Heisenberg and on the Engel groups; nilpotent Lie groups of 2-step and 3-step, respectively. Here we provide a preliminary analysis of the structure and of the symbolic calculus with symbols parametrized by (λ, μ) on the Engel group, while for the case of the Heisenberg group we recall the analogous results on the λ-classes of symbols.
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Notes
- 1.
For smooth vectors X, Y in \(\mathbb {R}^n\), we define the Lie-bracket [X, Y ] := Y X − XY .
- 2.
For V, W spaces of vector fields, we denote by [V, W] the set {[v, w] : v ∈ V, w ∈ W}.
References
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Acknowledgements
We wish to thank Professor Michael Ruzhansky for the useful discussions and suggestions that helped to improve the present work.
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Chatzakou, M. (2020). On (λ, μ)-Classes on the Engel Group. In: Georgiev, V., Ozawa, T., Ruzhansky, M., Wirth, J. (eds) Advances in Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-58215-9_2
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DOI: https://doi.org/10.1007/978-3-030-58215-9_2
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