Gagliardo-nirenberg Polynomial Growth Lie Group Inequality

In this paper,we prove that Gagliardo-Nirenberg inequalities with a BMO term is still valid on the Lie groups with polynomial growth, in order to do this,in the first part we re-call that Sobolev-Poincare inequality on polynomial growth Lie group, then by the use of heat semigroup Ht generated by Sublaplacian operator,we review some properties of Sobolev ,Hardy and BMO spaces on the groups with polynomial growth in the second part.