The analysis carried out in this dissertation lies in two settings: the Euclidean setting and the non-commutative setting of two stratified Lie groups; namely the Engel and the Cartan group. In the non-commutative direction, following the theory developed by Fischer and Ruzhansky, we set out the prerequisites for the development of the global symbolic calculus in the setting of the below groups. The latter means that explicit formulas for the difference operators in both settings are given, and subsequently the S^m(G) classes of symbols can be concretely described. The group Fourier transform of the sub-Laplacian on the aforementioned groups boils down to an anharmonic oscillator on Rn. This, in turn, hands over the study of anharmonic oscillators, using the Weyl-Hormander calculus, aiming to obtain spectral properties for their negative powers in terms of Schatten-von Neumann classes. Additionally, the associated Weyl-Hormander classes of the degenerate harmonic oscillator are described. Developing the Fourier analysis associated with the eigenfunctions of the, regarded as prototype, anharmonic oscillator, we establish a Hausdorff-Young-Paley inequality, and obtain boundedness results for Fourier multipliers in this setting. In the same spirit, but moving to the non-commutative setting, we apply a generalisation of a classical condition of Hormander on the Lp-Lq boundedness of Fourier multipliers on a locally compact group to spectral multipliers of non-Rockland operators on the Engel and Cartan groups.
Conditions on the r-nuclearity of functions of the prototype anharmonic oscillator
on modulation spaces are given, and these imply that Lidskii's trace formula that holds true for r >2/3 due to Grothendieck's theory, still holds true for larger values of r. Finally, we prove the Poincare inequality for a family of probability measures, with density depending on a homogeneous norm, on a class of stratifed Lie groups of
any step