  On the framework of the 2-adic group 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 Besov
space B(-1,\infty,\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 B(1,\infty,1). This
identification lead us to the construction of a counterexample to the improved
Sobolev inequality.
