A second look of Sobolev spaces on metrizable groups

We continue our study of Sobolev spaces on locally compact abelian groups. In this paper we mainly focus on the case of metrizable groups. We show the density of the Bruhat–Schwartz space in Sobolev space. We prove the trace theorem on the cartesian product of topological groups. The comparison of Sobolev and fractional Sobolev spaces are given. In particular, it is proved that in the case of any abelian connected Lie group Sobolev and fractional Sobolev spaces coincide. Most of the theorems are illustrated by p-adic groups.