This project will be consecrated to the investigation in various independent and intertwinedfields in analysis, the theory of pseudodifferential operators, harmonic analysis and the theoryof operators ideals. We will apply the theory of pseudodifferential operators to study degenerate elliptic equations, degenerate hyperbolic operators, fractional powers of subelliptic operators, Sobolev estimates. In particular we will investigate regularity on Sobolev spaces, invertibilityand the Cauchy problem for degenerate hyperbolic equations. The study of degenerate equations is a field of intenssive research with important applications in physics and engineering. A second field of interest will be the study of pseudodifferential operators on compact Lie groups applying techniques of Weyl-Hörmander calculus. Concerning nuclear operators and Schatten vonNeumann ideals we shall be interested in finding sufficient and/or neccesary conditionsfor the belongness to a such kind of ideals, in particular we will study different ideals of operatorson certain Lie groups and investigate the case of pseudodifferential operators. The study of traces is important in its own, traces of pseudodifferential operators play an essential role in the study of geometric and topological invariants. The belongness of a pseudodifferential operator to a Schatten-von Neumann ideal constitutes a way to measure its regularity, the case of localization operators is relevant in time-frequency analysis.