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Quantizations on the Engel and the Cartan groups

This work aims to develop a global quantization in the concrete settings of two graded nilpotent Lie groups of 3-step; namely of the Engel group and the Cartan group. We provide a preliminary analysis on the structure and the representations of the aforementioned groups, and their corresponding Lie algebras. In addition, the explicit formulas for the difference operators in the two settings are derived, constituting the necessary prerequisites for the constructions of the $Ψ^{m}_{ρ,δ}$ classes of symbols in both cases. In the case of the Engel group, the relation between the Kohn-Nirenberg quantization and the representations of the Engel group enables us to express operators in this setting in terms of quantization of symbols in the Euclidean space. We illustrate the $L^p-L^q$ boundedness for spectral multipliers theorems with particular examples of operators in both groups, that yield appropriate Sobolev-type inequalities. As a further application of the above, we get some consequences for the $L^p-L^q$ norm for the heat kernel of those particular examples of operators in both settings.