On the framework of the 2-adic group $mathcal{Z}_2$, we study a Sobolev-like inequality where we estimate the $L^2$ norm by a geometric mean of the BV norm and the $dot{B}_{infty}^{-1,{infty}}$ norm. We first show, using the special topological properties of the $p$-adic groups, that the set of functions of bounded variations BV can be identified to the Besov space ˙$dot{B}_1^{1,{infty}}$. This identification lead us to the construction of a counterexample to the improved Sobolev inequality.