Abstract
This chapter is meant to give a brief and by no means complete description of the Heisenberg group H\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {H}$$\end{document}, that will be the setting of this work. Customarily this group is presented as a particular group on R3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {R}^3$$\end{document}. This is not restrictive and to explain why we recall some definitions and basic properties of Carnot groups in order to make the exposition self-contained. We refer to the monograph (Bonfiglioli et al., Stratified Lie Groups and Potential Theory for their Sub-Laplacians, 2007 [1]) for a complete presentation of Carnot groups.