Paraproduct on the Heisenberg group and applications

We adapt the homogeneous Littlewood-Paley decomposition on the Heisenberg g roup constructed by H. Bahouri, P. Gerard and C.-J. Xu in [4] to the inhomo geneous case, which enables us to build paraproduct operators, similar to t hose defined by J.-M. Bony in [5]; although there is no simple formula for the Fourier transform of the product of two functions, some spectral locali zation properties of the classical case are preserved on the Heisenberg gro up after the product has been taken. Using the dyadic decomposition and the paraproduct algorithm, we prove the Gagliardo-Nirenberg inequality on the Heisenberg group; the smoothness of solutions of subelliptic, semi-linear s ystems is also studied, as well as semi-linear wave equations.