On $(\lambda,\mu)$-classes on the Engel group

TL;DR

This paper compares the symbolic pseudo-differential calculus on the Engel group with that on the Heisenberg group, analyzing the structure and properties of $(,)$-classes of symbols on the Engel group.

Contribution

It provides a preliminary analysis of the symbolic calculus with $(,)$-parametrized symbols on the Engel group, extending known results from the Heisenberg group.

Findings

Analysis of the structure of $(,)$-symbol classes on the Engel group

Comparison with $$-classes on the Heisenberg group

Insights into the properties of pseudo-differential calculus on nilpotent Lie groups

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 $(\lambda,\mu)$ on the Engel group, while for the case of the Heisenberg group we recall the analogous results on the $\lambda$-classes of symbols.