In a recent monograph entitled «Fourier Analysis and Nonlinear Partial Differential Equations», we gave the state-of-the-art of Fourier analysis methods in connection with partial differential equations in the whole space.

That book, which was intended to be elementary, encountered a great success among mathematicians. It motivated us to propose a similar monograph dedicated to a non Euclidean framework. As a start, we focus on the Heisenberg group which is the «simplest» noncommutative and infinite group on which a «reasonable» Fourier transform may be defined.