  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 $\Psi^{m}_{\rho,\delta}$ 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.
