  The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 <
p < +$\infty$. In the case p = 1 we study here a relaxed version of refined
Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces
associated to a weight from the Arino-Muckenhoupt class Bp, we will study
Gagliardo-Nirenberg inequalities. As a by-product we will also consider
Morrey-Sobolev inequalities. These arguments can be generalized to many
different frameworks, in particular the proofs are given in the setting of
stratified Lie groups.
