  We study in this article the Improved Sobolev inequalities with Muckenhoupt
weights within the framework of stratified Lie groups. This family of
inequalities estimate the Lq norm of a function by the geometric mean of two
norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty,
infty). When the value p which characterizes Sobolev space is strictly larger
than 1, the required result is well known in R^n and is classically obtained by
a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will
develop here another totally different technique. When p = 1, these two
techniques are not available anymore and following M. Ledoux in R^n, we will
treat here the critical case p = 1 for general stratified Lie groups in a
weighted functional space setting. Finally, we will go a step further with a
new generalization of Improved Sobolev inequalities using weak-type Sobolev
spaces.
